Reduce each of the following expressions to the sine and cosine of a single expression:

Solution:

(i) Let $f(x)=\sqrt{3} \sin x-\cos x$

Dividing and multiplying by $\sqrt{3+1}$, i. e. by 2 , we get :

$f(x)=2\left(\frac{\sqrt{3}}{2} \sin x-\frac{1}{2} \cos x\right)$

$\Rightarrow f(x)=2\left(\cos \frac{\pi}{6} \sin x-\sin \frac{\pi}{6} \cos x\right)$

$\Rightarrow f(x)=2 \sin \left(x-\frac{\pi}{6}\right)$

Again,

$f(x)=2\left(\frac{\sqrt{3}}{2} \sin x-\frac{1}{2} \cos x\right)$

$\Rightarrow f(x)=2\left(\sin \frac{\pi}{3} \sin x-\cos \frac{\pi}{3} \cos x\right)$

$\Rightarrow f(x)=-2 \cos \left(\frac{\pi}{3}+x\right)$

(ii) Let $f(x)=\cos x-\sin x$

Dividing and multiplying by $\sqrt{1^{2}+1^{2}}$, i.e. by $\sqrt{2}$, we get :

$f(x)=\sqrt{2}\left(\frac{1}{\sqrt{2}} \cos x-\frac{1}{\sqrt{2}} \sin x\right)$

$\Rightarrow f(x)=\sqrt{2}\left(\cos 45^{\circ} \cos x-\sin 45^{\circ} \sin x\right)$

$\Rightarrow f(x)=\sqrt{2} \cos \left(\frac{\pi}{4}+x\right)$

Again,

$f(x)=\sqrt{2}\left(\frac{1}{\sqrt{2}} \cos x-\frac{1}{\sqrt{2}} \sin x\right)$

$\Rightarrow f(x)=\sqrt{2}\left(\sin 45^{\circ} \cos x-\cos 45^{\circ} \sin x\right)$

$\Rightarrow f(x)=\sqrt{2} \sin \left(\frac{\pi}{4}-x\right)$

(iii) Let $f(x)=24 \cos x+7 \sin x$

Dividing and multiplying by $\sqrt{24^{2}+7^{2}}$, i.e. by 25, we get:

$f(x)=25\left(\frac{24}{25} \cos x+\frac{7}{25} \sin x\right)$

$\Rightarrow f(x)=25(\sin \alpha \cos x+\cos \alpha \sin x)$, where $\sin \alpha=\frac{24}{25}$ and $\cos \alpha=\frac{7}{25}$

$\Rightarrow f(x)=25 \sin (\alpha+x)$, where $\tan \alpha=\frac{24}{7}$

Again,

$f(x)=25\left(\frac{24}{25} \cos x+\frac{7}{25} \sin x\right)$

$\Rightarrow f(x)=25(\cos \alpha \cos x+\sin \alpha \sin x)$, where $\cos \alpha=\frac{24}{25}, \sin \alpha=\frac{7}{25} .$

$\Rightarrow f(x)=25 \cos (\alpha-x)$, where $\tan \alpha=\frac{7}{24} .$