Question

Determine whether the following equation is identity or not:

$\frac{\mathrm{cos\theta }}{1-\mathrm{tan\theta }}+\frac{\mathrm{sin\theta }}{1-\mathrm{cos\theta }}=\mathrm{sin\theta }+\mathrm{cos\theta }$.

Answer 1

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

$\frac{\mathrm{cos\theta }}{1-\mathrm{tan\theta }}+\frac{\mathrm{sin\theta }}{1-\mathrm{cos\theta }}=\mathrm{sin\theta }+\mathrm{cos\theta }$

Let, $\mathrm{L}.\mathrm{H}.\mathrm{S}.=\frac{\mathrm{cos\theta }}{1-\mathrm{tan\theta }}+\frac{\mathrm{sin\theta }}{1-\mathrm{cos\theta }}$

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

$\mathrm{L}.\mathrm{H}.\mathrm{S}.=\frac{\cos {30}^{\circ }}{1-\tan {30}^{\circ }}+\frac{\sin {30}^{\circ }}{1-\cos {30}^{\circ }}$

$$\begin{gathered} =\frac{{\displaystyle \frac{\sqrt{3}}{2}}}{1-{\displaystyle \frac{1}{\sqrt{3}}}}+\frac{{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{\left(\sqrt{3}-1\right)}+\frac{1}{2}\times \frac{2}{2-\sqrt{3}} \\ =\frac{3}{2\left(\sqrt{3}-1\right)}+\frac{1}{2-\sqrt{3}} \end{gathered}$$

$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\sin {30}^{\circ }+\cos {30}^{\circ }$

$$\begin{gathered} =\frac{1}{2}\times \frac{\sqrt{3}}{2} \\ =\frac{\sqrt{3}+1}{2} \end{gathered}$$

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

Hence, the given equation is not an identity.