(a) If sin θ+ cos θ=√ 3, then find the value of sin θ · cos θ.

\(\displaystyle \sin \theta+\cos \theta=\sqrt{3}\) squaring both sides

\[\begin{aligned} & \sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=3 \\ & \Rightarrow 1+2 \sin \theta \cos \theta=3 \\ & \Rightarrow \sin \theta \cos \theta=1 \end{aligned} \]

\(\displaystyle \operatorname{cosec} \alpha=\frac{1}{\sin \alpha}=\sqrt{2}\)

\(\displaystyle \operatorname{cosec} \beta=\sqrt{1+\cot ^{2} \beta}=\sqrt{1+3}=2\)

\(\displaystyle \therefore \operatorname{cosec} \alpha+\operatorname{cosec} \beta=\sqrt{2}+2\) or \(\displaystyle \sqrt{2}(\sqrt{2}+1)\)