Question

$\left\{\frac{1}{({\sec }^2\mathrm{\theta }-{\cos }^2\mathrm{\theta })}+\frac{1}{({\mathrm{cosec}}^2\mathrm{\theta }-{\sin }^2\mathrm{\theta })}\right\}({\sin }^2\mathrm{\theta }{\cos }^2\mathrm{\theta })=\frac{1-{\sin }^2{\mathrm{\theta cos}}^2\mathrm{\theta }}{2+{\sin }^2\mathrm{\theta }{\cos }^2\mathrm{\theta }}$

Answer 1

$$\begin{gathered} \begin{array}{l}\\ \text{LHS=}\{\frac{1}{{\text{sec}}^2\theta -{\text{cos}}^2\theta }+\frac{1}{{\text{cosec}}^2\theta -{\text{sin}}^2\theta }\}\left({\text{sin}}^2\theta {\text{cos}}^2\theta \right)\\ \text{=}\{\frac{{\text{cos}}^2\theta }{1-{\text{cos}}^4\theta }+\frac{{\text{sin}}^2\theta }{1-{\text{sin}}^4\theta }\}\left({\text{sin}}^2\theta {\text{cos}}^2\theta \right)\\ \text{=}\{\frac{{\text{cos}}^2\theta }{(1-{\text{cos}}^2\theta )\left({\text{1+cos}}^2\theta \right)}+\frac{{\sin }^2\theta }{(1-{\sin }^2\theta )(1+{\sin }^2\theta )}\}\left({\text{sin}}^2\theta {\text{cos}}^2\theta \right)\\ \text{=}[\frac{{\cot }^2\theta }{1+{\cos }^2\theta }+\frac{{\tan }^2\theta }{1+{\sin }^2\theta }]{\sin }^2\theta {\cos }^2\theta \\ \text{=}\frac{{\cos }^4\theta }{1+{\cos }^2\theta }+\frac{{\sin }^4\theta }{1+{\sin }^2\theta }\\ \text{=}\frac{{\left({\cos }^2\theta \right)}^2}{1+{\cos }^2\theta }+\frac{{\left({\sin }^2\theta \right)}^2}{1+{\sin }^2\theta }\\ \text{=}\frac{(1-{\sin }^2\theta )}{1+{\cos }^2\theta }+\frac{{(1-{\cos }^2\theta )}^2}{1+{\sin }^2\theta }\\ \text{=}\frac{{(1-{\sin }^2\theta )}^2(1+{\sin }^2)+{(1-{\cos }^2\theta )}^2(1+{\text{cos}}^2\theta )}{(1+{\sin }^2\theta )(1+{\text{cos}}^2\theta )}\\ \text{=}\frac{{\cos }^4\theta (1+{\sin }^2\theta )+{\sin }^4\theta (1+{\text{cos}}^2\theta )}{1+{\sin }^2\theta +{\cos }^2\theta +{\sin }^2\theta {\cos }^2\theta }\end{array} \\ =\frac{{\cos }^4\theta +{\cos }^4\theta {\sin }^2\theta +{\sin }^4\theta +{\sin }^4\theta {\text{cos}}^2\theta }{1+1+{\sin }^2\theta {\cos }^2\theta } \\ =\frac{{\cos }^4\theta +{\sin }^4\theta +{\sin }^2\theta {\cos }^2\theta ({\sin }^2\theta +{\text{cos}}^2\theta )}{2+{\sin }^2\theta {\cos }^2\theta } \\ =\frac{({\cos }^2\theta {)}^2+({\sin }^2\theta {)}^2+{\sin }^2\theta {\cos }^2\theta \left(1\right)}{2+{\sin }^2\theta {\cos }^2\theta } \\ =\frac{({\cos }^2\theta +{\sin }^2\theta {)}^2-2{\sin }^2\theta {\cos }^2\theta +{\sin }^2\theta {\cos }^2\theta \left(1\right)}{2+{\sin }^2\theta {\cos }^2\theta } \\ \begin{array}{l}\text{=}\frac{1^2+{\cos }^2\theta {\sin }^2\theta -2{\cos }^2\theta {\sin }^2\theta }{2+{\sin }^2\theta {\cos }^2\theta }\\ \text{=}\frac{1-{\cos }^2\theta {\sin }^2\theta }{2+{\sin }^2\theta {\cos }^2\theta }\end{array} \\ =\mathrm{RHS} \end{gathered}$$