Question

Determine whether the following equation is identity or not:

$\mathrm{cot\theta }+\mathrm{tan\theta }=\frac{\mathrm{sec\theta }}{\mathrm{sin\theta }}$.

Answer 1

Verify L.H.S and R.H.S at any value of $\mathit{\theta }$

$\mathrm{cot\theta }+\mathrm{tan\theta }=\frac{\mathrm{sec\theta }}{\mathrm{sin\theta }}$

Let, $\mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{cot\theta }+\mathrm{tan\theta }$

when $\mathrm{\theta }={30}^{\circ }$

$\mathrm{L}.\mathrm{H}.\mathrm{S}.=\cot {30}^{\circ }+\tan {30}^{\circ }$

$$\begin{gathered} =\sqrt{3}+\frac{1}{\sqrt{3}} \\ =\frac{3+1}{\sqrt{3}} \\ =\frac{4}{\sqrt{3}} \end{gathered}$$

$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{\sec \theta }{\sin \theta }$

$$\begin{gathered} =\frac{sec{30}^{\circ }}{\sin {30}^{\circ }} \\ =\frac{2}{\sqrt{3}}\times 2 \\ =\frac{4}{\sqrt{3}} \end{gathered}$$

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

Hence, the given equation is an identity.