Prove that: $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$
$L.H.S.$ $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}$
$=\cos ^{2} x+\cos ^{2} y-2 \cos x \cos y+\sin ^{2} x+\sin ^{2} y-2 \sin x \sin y$
$=\left(\cos ^{2} x+\sin ^{2} x\right)+\left(\cos ^{2} y+\sin ^{2} y\right)-2[\cos x \cos y+\sin x \sin y]$
$=1+1-2[\cos (x-y)]$
$[\cos (A-B)=\cos A \cos B+\sin A \sin B]$
$=2[1-\cos (x-y)]$
$=2\left[1-\left\{1-2 \sin ^{2}\left(\frac{x-y}{2}\right)\right\}\right] \quad\left[\cos 2 A=1-2 \sin ^{2} A\right]$
$=4 \sin ^{2}\left(\frac{x-y}{2}\right)= R . H.S.$
Prove that:
$2 \sin ^{2} \frac{\pi}{6}+\cos ec ^{2} \frac{7 \pi}{6} \cos ^{2} \frac{\pi}{3}=\frac{3}{2}$
If $\sin \theta = \frac{{ - 4}}{5}$ and $\theta $ lies in the third quadrant, then $\cos \frac{\theta }{2} = $
Find the values of other five trigonometric functions if $\sec x=\frac{13}{5}, x$ lies in fourth quadrant.
If ${\tan ^2}\alpha {\tan ^2}\beta + {\tan ^2}\beta {\tan ^2}\gamma + {\tan ^2}\gamma {\tan ^2}\alpha $$ + 2{\tan ^2}\alpha {\tan ^2}\beta {\tan ^2}\gamma = 1,$ then the value of ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma $ is
The value of $\cot \frac{\pi}{24}$ is :