सिद्ध कीजिए
$(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$
सिद्ध कीजिए
$(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$
$L.H.S.$ $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}$
$=\cos ^{2} x+\cos ^{2} y-2 \cos x \cos y+\sin ^{2} x+\sin ^{2} y-2 \sin x \sin y$
$=\left(\cos ^{2} x+\sin ^{2} x\right)+\left(\cos ^{2} y+\sin ^{2} y\right)-2[\cos x \cos y+\sin x \sin y]$
$=1+1-2[\cos (x-y)]$
$[\cos (A-B)=\cos A \cos B+\sin A \sin B]$
$=2[1-\cos (x-y)]$
$=2\left[1-\left\{1-2 \sin ^{2}\left(\frac{x-y}{2}\right)\right\}\right] \quad\left[\cos 2 A=1-2 \sin ^{2} A\right]$
$=4 \sin ^{2}\left(\frac{x-y}{2}\right)= R . H.S.$
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
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
$2 \sin ^{2} \frac{\pi}{6}+\operatorname{cosec}^{2} \frac{7 \pi}{6} \cos ^{2} \frac{\pi}{3}=\frac{3}{2}$
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
यदि $A + B + C = \pi $ तथा $\cos A = \cos B\,\cos C,$ तब $\tan B\,\,\tan C$ का मान होगा
सिद्ध कीजिएः
$\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}=-\frac{1}{2}$
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
यदि $\cos \theta - \sin \theta = \sqrt 2 \sin \theta ,$ तो $\cos \theta + \sin \theta $ बराबर होगा