==>${\tan ^{ - 1}}\left[ {\frac{{\frac{1}{{\sqrt {\cos \alpha } }} - \sqrt {\cos \alpha } }}{{1 + \frac{{\sqrt {\cos \alpha } }}{{\sqrt {\cos \alpha } }}}}} \right] = x$
==> $\tan x = \frac{{1 - \cos \alpha }}{{2\sqrt {\cos \alpha } }}$
$\therefore \sin x = \frac{{1 - \cos \alpha }}{{1 + \cos \alpha }} = \frac{{2{{\sin }^2}\frac{\alpha }{2}}}{{2{{\cos }^2}\frac{\alpha }{2}}} = {\tan ^2}\left( {\frac{\alpha }{2}} \right)$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Let $x _1< x _2< x _3<\ldots< x _{ n }<\ldots$ be all the points of local maximum of $f$ and $y_1$
$(1)$ $\left|x_n-y_n\right|>1$ for every $n$
$(2)$ $x_1 < y _1$
$(3)$ $x_n \in\left(2 n , 2 n +\frac{1}{2}\right)$ for every $n$
$(4)$ $x_{n+1}-x_n>2$ for every $n$
$(S1)$: $f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$
$( S 2): \int_{-2}^{2} f ( x ) dx =12$Then,
$ 3 x+5 y+\lambda z=3 $
$ 7 x+11 y-9 z=2 $
$ 97 x+155 y-189 z=\mu$
has infinitely many solutions, then $\mu+2 \lambda$ is equal to :