Question

prove that √1/sin²Theta -sin²theta -cos²theta =cot theta​

Answer 1

$\underline{\underline{\bf{ prove \: that }}} : - \\ \\ \sqrt{ \frac{1}{ { \sin }^{2} \theta } - { \sin }^{2} \theta - { \cos }^{2} \theta } = \cot \theta \\ \\ \underline{ \underline{\bf{{step - by \: step \: explanation}}}} : -$

Firstly,take Left hand side (LHL)

$\rightarrow \: \sqrt{ \frac{1}{ { \sin}^{2} \theta } - { \sin }^{2} \theta - { \cos }^{2} \theta } \\ \\ \rightarrow \: \sqrt{ \frac{1}{ { \sin}^{2} \theta} - ( { \sin }^{2} \theta + { \cos }^{2} \theta) } \\ \\ \because \: { \sin}^{2} \theta + { \cos }^{2} \theta = 1 \\ \\ \therefore \\ \\ \rightarrow \: \sqrt{ \frac{1}{ { \sin }^{2} \theta} - 1} \\ \\ \rightarrow \: \sqrt{ \frac{1 - { \sin }^{2} \theta}{ { \sin }^{2} \theta } } \\ \\ \because \: 1 - { \sin }^{2} \theta \: = { \cos }^{2} \theta \\ \\ \therefore \: \\ \\ \rightarrow \sqrt{ \frac{ { \cos}^{2} \theta}{ { \sin }^{2} \theta} } \\ \\ \rightarrow \: \sqrt{ { \cot}^{2} \theta} \\ \\ \rightarrow \: \cot \theta$

Left hand side = Right hand side ,

Hence proved.