(a) মান নির্নয় করো :   $\cos{e} c \theta + \cot{\theta} = \sqrt{3}$  

(b) যদি   $\sin^{- 1}{x} + \sin^{- 1}{y} = \frac{\pi}{2}$ হয় , তাহলে দেখাও যে x²+y²=1

a)

$\operatorname{cosec} \theta+\cot \theta=\sqrt{3}$

$\begin{array}{l}\Rightarrow \frac{1+\cos \theta}{\sin \theta}=\sqrt{3} \Rightarrow \frac{2 \cos ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}}=\sqrt{3} \\\Rightarrow \cos \frac{\theta}{2}\left(\cos \frac{\theta}{2}-\sqrt{3} \sin \frac{\theta}{2}\right)=0\end{array}$

হয়, $\cos \frac{\theta}{2}-\sqrt{3} \sin \frac{\theta}{2}=0 \quad$ অথবা, $\cos \frac{\theta}{2}=0$

$\Rightarrow \cot \theta / 2=\sqrt{3} \quad \therefore \theta=(2 n+1) \pi$

$\Rightarrow \tan \frac{\theta}{2}=\frac{1}{\sqrt{3}}=\tan \frac{\pi}{6}$

$\therefore \frac{\theta}{2}=\mathrm{n} \pi+\frac{\pi}{6} \therefore \theta=2 \mathrm{n} \pi+\frac{\pi}{3}$

b)

$\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{2}$

$\begin{array}{l}\Rightarrow x=\sin \left(\frac{\pi}{2}-\sin ^{-1} y\right)=\cos \left(\sin ^{-1} y\right) \\=\cos \left(\cos ^{-1} \sqrt{1-y^{2}}\right) \\\Rightarrow x=\sqrt{1-y^{2}} \cdot \therefore x^{2}+y^{2}=1 \text { Proved. } \end{array}$