# Write the value of cos

Question:

Write the value of $\cos ^{2} 76^{\circ}+\cos ^{2} 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ} .$

Solution:

We have,

$\cos ^{2} 76^{\circ}+\cos ^{2} 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ}$

$=\frac{1}{2}\left[1+\cos 2(76)^{\circ}+1+\cos 2(16)^{\circ}-\cos (76+16)^{\circ}-\cos (76-16)^{\circ}\right]$

$\left[\because 2 \cos ^{2} \theta=1+\cos 2 \theta\right.$ and $\left.2 \cos A \cos B=\cos (A+B)+\cos (A-B)\right]$

$=\frac{1}{2}\left[2+\cos 152^{\circ}+\cos 32^{\circ}-\cos 92^{\circ}-\frac{1}{2}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos \left(180-152^{\circ}\right)+\cos 32^{\circ}-\cos 92^{\circ}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+2 \sin \frac{92^{\circ}+32^{\circ}}{2} \sin \frac{92^{\circ}-32^{\circ}}{2}\right]$

$\left[\cos C-\cos D=2 \sin \frac{C+D}{2} \sin \frac{D-C}{2}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+2 \sin \frac{124^{\circ}}{2} \sin \frac{60^{\circ}}{2}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+2 \sin 62^{\circ} \sin 30^{\circ}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+2 \sin 62^{\circ} \times \frac{1}{2}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+\sin 62^{\circ}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+\sin (90-28)^{\circ}\right]$

$=\frac{1}{2}\left[\frac{3}{2}-\cos 28^{\circ}+\cos 28^{\circ}\right]$

$=\frac{3}{4}$