Question

Prove the given identity by using the identity $1+{\cos }^2\theta =\cos e{c}^2\theta$:

$1+\frac{{\cot }^2\theta }{1+\mathrm{cosec\theta }}=\mathrm{cosec\theta }$.

Answer 1

Verify L.H.S and R.H.S

Let $\mathrm{L}.\mathrm{H}.\mathrm{S}.=1+\frac{{\cot }^2\mathrm{\theta }}{1+\mathrm{cosec\theta }}$

$$\begin{gathered} =1+\frac{{\mathrm{cosec}}^2\mathrm{\theta }-1}{1+\mathrm{cosec\theta }} \\ =1+\mathrm{cosec\theta }-1 \\ =\mathrm{cosec\theta } \\ =\mathrm{R}.\mathrm{H}.\mathrm{S}. \end{gathered}$$

Hence, $\mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{R}.\mathrm{H}.\mathrm{S}.$