સાબિત કરો કે, $\cos ^{2} x+\cos ^{2}\left(x+\frac{\pi}{3}\right)+\cos ^{2}\left(x-\frac{\pi}{3}\right)=\frac{3}{2}$
We have
${\text{L}}{\text{.H}}{\text{.S}}{\text{.}}$
$ = \frac{{1 + \cos 2x}}{2} + \frac{{1 + \cos \left( {2x + \frac{{2\pi }}{3}} \right)}}{2} + \frac{{1 + \cos \left( {2x - \frac{{2\pi }}{3}} \right)}}{2}$
$ = \frac{1}{2}\left[ {3 + \cos 2x + \cos \left( {2x + \frac{{2\pi }}{3}} \right) + \cos \left( {2x - \frac{{2\pi }}{3}} \right)} \right]$
$ = \frac{1}{2}\left[ {3 + \cos 2x + 2\cos 2x\cos \frac{{2\pi }}{3}} \right]$
$=\frac{1}{2}\left[3+\cos 2 x+2 \cos 2 x \cos \left(\pi-\frac{\pi}{3}\right)\right]$
$=\frac{1}{2}\left[3+\cos 2 x-2 \cos 2 x \cos \frac{\pi}{3}\right]$
$=\frac{1}{2}[3+\cos 2 x-\cos 2 x]=\frac{3}{2}= R.H.S.$
જો $a\,{\cos ^3}\alpha + 3a\,\cos \alpha \,{\sin ^2}\alpha = m$ અને $a\,{\sin ^3}\alpha + 3a\,{\cos ^2}\alpha \sin \alpha = n,$ તો ${(m + n)^{2/3}} + {(m - n)^{2/3}} = . . .$
$\sin \frac{x}{2}, \cos \frac{x}{2}$ અને $\tan \frac{x}{2}$ ની કિંમતો શોધો.: $\cos x=-\frac{1}{3}, x$ એ બીજા ચરણમાં છે.
જો $\tan \theta - \cot \theta = a$ અને $\sin \theta + \cos \theta = b,$ તો ${({b^2} - 1)^2}({a^2} + 4)$ મેળવો.
$\sin \frac{31 \pi}{3}$ નું મૂલ્ય શોધો.
રેડિયન માપ શોધો : $25^{\circ}$