सिद्ध कीजिए

$(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$

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$L.H.S.$ $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}$

$=\cos ^{2} x+\cos ^{2} y-2 \cos x \cos y+\sin ^{2} x+\sin ^{2} y-2 \sin x \sin y$

$=\left(\cos ^{2} x+\sin ^{2} x\right)+\left(\cos ^{2} y+\sin ^{2} y\right)-2[\cos x \cos y+\sin x \sin y]$

$=1+1-2[\cos (x-y)]$

$[\cos (A-B)=\cos A \cos B+\sin A \sin B]$

$=2[1-\cos (x-y)]$

$=2\left[1-\left\{1-2 \sin ^{2}\left(\frac{x-y}{2}\right)\right\}\right] \quad\left[\cos 2 A=1-2 \sin ^{2} A\right]$

$=4 \sin ^{2}\left(\frac{x-y}{2}\right)= R . H.S.$

Similar Questions

यदि ${\tan ^2}\alpha \;{\tan ^2}\beta  + {\tan ^2}\beta \;{\tan ^2}\gamma  + {\tan ^2}\gamma \;{\tan ^2}\alpha  + 2{\tan ^2}\alpha \;{\tan ^2}\beta \;{\tan ^2}\gamma  = 1,$ तब

${\sin ^2}\alpha  + {\sin ^2}\beta  + {\sin ^2}\gamma $ का मान है

सिद्ध कीजिएः

$2 \sin ^{2} \frac{\pi}{6}+\operatorname{cosec}^{2} \frac{7 \pi}{6} \cos ^{2} \frac{\pi}{3}=\frac{3}{2}$

यदि $A + B + C = \pi $ तथा $\cos A = \cos B\,\cos C,$ तब $\tan B\,\,\tan C$ का मान होगा

सिद्ध कीजिएः

$\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}=-\frac{1}{2}$

यदि $\cos \theta - \sin \theta = \sqrt 2 \sin \theta ,$ तो $\cos \theta + \sin \theta $ बराबर होगा