Prove that:

Solution:

(i) LHS $=\cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ}$

$=\frac{1}{2}\left[2 \cos 10^{\circ} \cos 50^{\circ}\right] \cos 30^{\circ} \cos 70^{\circ}$

$=\frac{1}{2}\left[\cos \left(10^{\circ}+50^{\circ}\right)+\cos \left(10^{\circ}-50^{\circ}\right)\right] \cos 30^{\circ} \cos 70^{\circ}$

$\{\because 2 \cos A \cos B=\cos (A+B)-\cos (A-B)\}$

$=\frac{1}{2}\left[\cos 60^{\circ}+\cos \left(-40^{\circ}\right)\right] \cos 30^{\circ} \cos 70^{\circ}$

$=\frac{1}{2}\left[\frac{1}{2}+\cos 40^{\circ}\right]\left(\frac{\sqrt{3}}{2}\right) \times \cos 70^{\circ}$

$=\frac{\sqrt{3}}{4} \cos 70^{\circ}\left[\frac{1}{2}+\cos 40^{\circ}\right]$

$=\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{\sqrt{3}}{4}\left[\cos 70^{\circ} \cos 40^{\circ}\right]$

$=\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{\sqrt{3}}{8}\left[2 \cos 70^{\circ} \cos 40^{\circ}\right]$

$=\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{\sqrt{3}}{8}\left[\cos \left(70^{\circ}+40^{\circ}\right)+\cos \left(70^{\circ}-40^{\circ}\right)\right]$

$=\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{\sqrt{3}}{8}\left[\cos 110^{\circ}+\cos 30^{\circ}\right]$

$=\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{\sqrt{3}}{8}\left[\cos \left(180^{\circ}-70^{\circ}\right)+\frac{\sqrt{3}}{2}\right]$

$=\frac{\sqrt{3}}{2} \cos 70^{\circ}-\frac{\sqrt{3}}{8} \cos 70^{\circ}+\frac{3}{16} \quad\left[\because \cos \left(180^{\circ}-70^{\circ}\right)=-\cos 70^{\circ}\right]$

$=\frac{3}{16}=\mathrm{RHS}$

(ii) $\mathrm{LHS}=\cos 40^{\circ} \cos 80^{\circ} \cos 160^{\circ}$

$=\frac{1}{2}\left[2 \cos 40^{\circ} \cos 80^{\circ}\right] \cos 160^{\circ}$

$=\frac{1}{2}\left[\cos \left(40^{\circ}+80^{\circ}\right)+\cos \left(40^{\circ}-80^{\circ}\right)\right] \cos 160^{\circ}$

$=\frac{1}{2}\left[\cos 120^{\circ}+\cos \left(-40^{\circ}\right)\right] \cos 160^{\circ}$

$=\frac{1}{2} \cos \left(160^{\circ}\right)\left[-\frac{1}{2}+\cos 40^{\circ}\right]$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{2} \cos 160^{\circ} \cos 40^{\circ}$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{4}\left[2 \cos 160^{\circ} \cos 40^{\circ}\right]$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{4}\left[\cos \left(160^{\circ}+40^{\circ}\right)+\cos \left(160^{\circ}-40^{\circ}\right)\right]$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{4}\left[\cos 200^{\circ}+\cos 120^{\circ}\right]$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{4}\left[\cos \left(360^{\circ}-160^{\circ}\right)-\frac{1}{2}\right]$

$=-\frac{1}{4} \cos 160^{\circ}+\frac{1}{4} \cos 160^{\circ}-\frac{1}{8} \quad\left[\because \cos \left(360^{\circ}-160^{\circ}\right)=\cos 160^{\circ}\right]$

$=-\frac{1}{8}=\mathrm{RHS}$

(iii) LHS $=\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}$

$=\frac{1}{2}\left[2 \sin 20^{\circ} \sin 40^{\circ}\right] \sin 80^{\circ}$

$=\frac{1}{2}\left[\cos \left(20^{\circ}-40^{\circ}\right)-\cos \left(20^{\circ}+40^{\circ}\right)\right] \sin 80^{\circ}$

$=\frac{1}{2}\left[\cos 20^{\circ}-\frac{1}{2}\right] \sin 80^{\circ}$

$=\frac{1}{2} \sin 80^{\circ}\left[\cos 20^{\circ}-\frac{1}{2}\right]$

$=\frac{1}{2} \sin 80^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{2} \sin \left(90^{\circ}-10^{\circ}\right) \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{2} \cos 10^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{4}\left[2 \cos 10^{\circ} \cos 20^{\circ}\right]-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{4}\left[\cos \left(10^{\circ}+20^{\circ}\right)+\cos \left(10^{\circ}-20^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(-10^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}$

$=\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(90^{\circ}-80^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{8}+\frac{1}{4} \sin 80^{\circ}-\frac{1}{4} \sin 80^{\circ}$     $\left[\because \cos \left(90^{\circ}-80^{\circ}\right)=\sin 80^{\circ}\right]$

$=\frac{\sqrt{3}}{8}=\mathrm{RHS}$

(iv) LHS $=\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}$

$=\frac{1}{2}\left[2 \cos 20^{\circ} \cos 40^{\circ}\right] \cos 80^{\circ}$

$=\frac{1}{2}\left[\cos \left(20^{\circ}+40^{\circ}\right)+\cos \left(20^{\circ}-40^{\circ}\right)\right] \cos 80^{\circ}$

$=\frac{1}{2}\left[\cos 60^{\circ}+\cos \left(-20^{\circ}\right)\right] \cos 80^{\circ}$

$=\frac{1}{2} \cos 80^{\circ}\left[\frac{1}{2}+\cos 20^{\circ}\right]$

$=\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}$

$=\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[2 \cos 80^{\circ} \cos 20^{\circ}\right]$

$=\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(80^{\circ}+20^{\circ}\right)+\cos \left(80^{\circ}-20^{\circ}\right)\right]$

$=\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos 100^{\circ}+\cos 60^{\circ}\right]$

$=\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(180^{\circ}-80^{\circ}\right)+\frac{1}{2}\right]$

$=\frac{1}{4} \cos 80^{\circ}-\frac{1}{4} \cos 80^{\circ}+\frac{1}{8}$       $\left\{\because \cos \left(180^{\circ}-80^{\circ}\right)=-\cos 80^{\circ}\right\}$

$=\frac{1}{8}=\mathrm{RHS}$

(v) $1 \mathrm{HS}=\tan 20^{\circ} \tan 40^{\circ} \tan 60^{\circ} \tan 80^{\circ}$

$=\tan 60^{\circ} \frac{\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2}\left[2 \sin 20^{\circ} \sin 40^{\circ}\right] \sin 80^{\circ}}{\frac{1}{2}\left[2 \cos 20^{\circ} \cos 40^{\circ}\right] \cos 80^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2}\left[\cos \left(20^{\circ}-40^{\circ}\right)-\cos \left(20^{\circ}+40^{\circ}\right)\right] \sin 80^{\circ}}{\frac{1}{2}\left[\cos \left(20^{\circ}+40^{\circ}\right)+\cos \left(20^{\circ}-40^{\circ}\right)\right] \cos 80^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2}\left[\cos \left(-20^{\circ}\right)-\cos 60^{\circ}\right] \sin 80^{\circ}}{\frac{1}{2}\left[\cos 60^{\circ}+\cos \left(-20^{\circ}\right)\right] \cos 80^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2} \sin 80^{\circ}\left[\cos 20^{\circ}-\frac{1}{2}\right]}{\frac{1}{2} \cos 80^{\circ}\left[\frac{1}{2}+\cos 20^{\circ}\right]}$

$=\sqrt{3} \times \frac{\frac{1}{2} \sin 80^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2} \sin \left(90^{\circ}-10^{\circ}\right) \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{2} \cos 10^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[2 \cos 10^{\circ} \cos 20^{\circ}\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[2 \cos 80^{\circ} \cos 20^{\circ}\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos \left(10^{\circ}+20^{\circ}\right)+\cos \left(10^{\circ}-20^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(80^{\circ}+20^{\circ}\right)+\cos \left(80^{\circ}-20^{\circ}\right)\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(-10^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos 100^{\circ}+\cos 60^{\circ}\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(90^{\circ}-80^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(180^{\circ}-80^{\circ}\right)+\frac{1}{2}\right]}$

$=\sqrt{3} \times \frac{\frac{\sqrt{3}}{8}+\frac{1}{4} \sin 80^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}-\frac{1}{4} \cos 80^{\circ}+\frac{1}{8}}\left[\cos \left(90^{\circ}-80^{\circ}\right)=\sin 80^{\circ}\right.$, and $\left.\cos \left(180^{\circ}-80^{\circ}\right)=-\cos \left(80^{\circ}\right)\right]$

$=\sqrt{3} \times \frac{\frac{\sqrt{3}}{8}}{\frac{1}{8}}$

$=3=\mathrm{RHS}$

(vi) LHS = tan 20° tan 30° tan 60° tan 80°

$=\tan 30^{\circ} \frac{\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2}\left[2 \sin 20^{\circ} \sin 40^{\circ}\right] \sin 80^{\circ}}{\frac{1}{2}\left[2 \cos 20^{\circ} \cos 40^{\circ}\right] \cos 80^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2}\left[\cos \left(20^{\circ}-40^{\circ}\right)-\cos \left(20^{\circ}+40^{\circ}\right)\right] \sin 80^{\circ}}{\frac{1}{2}\left[\cos \left(20^{\circ}+40^{\circ}\right)+\cos \left(20^{\circ}-40^{\circ}\right)\right] \cos 80^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2}\left[\cos 20^{\circ}-\frac{1}{2}\right] \sin 80^{\circ}}{\frac{1}{2}\left[\cos 60^{\circ}+\cos \left(-20^{\circ}\right)\right] \cos 80^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2} \sin 80^{\circ}\left[\cos 20^{\circ}-\frac{1}{2}\right]}{\frac{1}{2} \cos 80^{\circ}\left[\frac{1}{2}+\cos 20^{\circ}\right]}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2} \sin 80^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2} \sin \left(90^{\circ}-10^{\circ}\right) \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\frac{1}{\sqrt{3}} \times \frac{\frac{1}{2} \cos 10^{\circ} \cos 20^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{2} \cos 80^{\circ} \cos 20^{\circ}}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[2 \cos 10^{\circ} \cos 20^{\circ}\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[2 \cos 80^{\circ} \cos 20^{\circ}\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos \left(10^{\circ}+20^{\circ}\right)+\cos \left(10^{\circ}-20^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(80^{\circ}+20^{\circ}\right)+\cos \left(80^{\circ}-20^{\circ}\right)\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(-10^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos 100^{\circ}+\cos 60^{\circ}\right]}$

$=\sqrt{3} \times \frac{\frac{1}{4}\left[\cos 30^{\circ}+\cos \left(90^{\circ}-80^{\circ}\right)\right]-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}+\frac{1}{4}\left[\cos \left(180^{\circ}-80^{\circ}\right)+\frac{1}{2}\right]}$

$=\sqrt{3} \times \frac{\frac{\sqrt{3}}{8}+\frac{1}{4} \sin 80^{\circ}-\frac{1}{4} \sin 80^{\circ}}{\frac{1}{4} \cos 80^{\circ}-\frac{1}{4} \cos \left(80^{\circ}\right)+\frac{1}{8}} \quad\left\{\because \cos \left(90^{\circ}-80^{\circ}\right)=\sin 80^{\circ}, \cos \left(180^{\circ}-80^{\circ}\right)=-\cos 80^{\circ}\right\}$

$=\sqrt{3} \times \frac{\frac{\sqrt{3}}{8}}{\frac{1}{8}}$

$=3=\mathrm{RHS}$

(vii) $\mathrm{LHS}=\sin 10^{\circ} \sin 50^{\circ} \sin 60^{\circ} \sin 70^{\circ}$

$=\frac{1}{2} \sin 60^{\circ}\left[2 \sin 10^{\circ} \sin 50^{\circ}\right] \sin 70^{\circ}$

$=\frac{1}{2} \times \frac{\sqrt{3}}{2}\left[\cos \left(10^{\circ}-50^{\circ}\right)-\cos \left(10^{\circ}+50^{\circ}\right)\right] \sin 70^{\circ}$

$=\frac{\sqrt{3}}{4}\left[\cos \left(-40^{\circ}\right)-\frac{1}{2}\right] \sin 70^{\circ}$

$=\frac{\sqrt{3}}{4} \sin 70^{\circ}\left[\cos 40^{\circ}-\frac{1}{2}\right]$

$=\frac{\sqrt{3}}{4} \sin 70^{\circ} \cos 40^{\circ}-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{4} \sin \left(90^{\circ}-20^{\circ}\right) \cos 40^{\circ}-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{4} \cos 20^{\circ} \cos 40^{\circ}-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{8}\left[2 \cos 20^{\circ} \cos 40^{\circ}\right]-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos \left(20^{\circ}+40^{\circ}\right)+\cos \left(20^{\circ}-40^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos 60^{\circ}+\cos \left(-20^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos 60^{\circ}+\cos \left(90^{\circ}-70^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 70^{\circ}$

$=\frac{\sqrt{3}}{16}+\frac{\sqrt{3}}{8} \sin 70^{\circ}-\frac{\sqrt{3}}{8} \sin 70^{\circ} \quad\left[\because \cos \left(90^{\circ}-70^{\circ}\right)=\sin 70^{\circ}\right]$

$=\frac{\sqrt{3}}{16}=$ RHS

(viii) LHS $=\sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}$

$=\frac{1}{2} \sin 60^{\circ}\left[2 \sin 20^{\circ} \sin 40^{\circ}\right] \sin 80^{\circ}$

$=\frac{1}{2} \times \frac{\sqrt{3}}{2}\left[\cos \left(20^{\circ}-40^{\circ}\right)-\cos \left(20^{\circ}+40^{\circ}\right)\right] \sin 80^{\circ}$

$=\frac{\sqrt{3}}{4}\left[\cos 20^{\circ}-\frac{1}{2}\right] \sin 80^{\circ}$

$=\frac{\sqrt{3}}{4} \sin 80^{\circ}\left[\cos 20^{\circ}-\frac{1}{2}\right]$

$=\frac{\sqrt{3}}{4} \sin 80^{\circ} \cos 20^{\circ}-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{4} \sin \left(90^{\circ}-10^{\circ}\right) \cos 20^{\circ}-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{4} \cos 10^{\circ} \cos 20^{\circ}-\frac{\sqrt{3}}{8} \sin \left(80^{\circ}\right)$

$=\frac{\sqrt{3}}{8}\left[2 \cos 10^{\circ} \cos 20^{\circ}\right]-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos \left(10^{\circ}+20^{\circ}\right)+\cos \left(10^{\circ}-20^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos 30^{\circ}+\cos \left(-10^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{\sqrt{3}}{8}\left[\cos 30^{\circ}+\cos \left(90^{\circ}-80^{\circ}\right)\right]-\frac{\sqrt{3}}{8} \sin 80^{\circ}$

$=\frac{3}{16}+\frac{\sqrt{3}}{8} \sin 80^{\circ}-\frac{\sqrt{3}}{8} \sin 80^{\circ} \quad\left[\because \cos \left(90^{\circ}-80^{\circ}\right)=\sin 80^{\circ}\right]$

$=\frac{3}{16}=$ RHS