Question

Prove that:

cosec A (1 − cos A) (cosec A + cot A) = 1

Answer 1

Solving L.H.S. of the equation:

\[\Rightarrow \operatorname{cosec} A (1 - \cos A)(\operatorname{cosec} A + \cot A)\]

\[\Rightarrow \frac{1}{\sin A} (1 - \cos A) \left(\frac{1}{\sin A} + \frac{\cos A}{\sin A}\right)\]

\[\Rightarrow \frac{1}{\sin A} (1 - \cos A) \times \frac{1 + \cos A}{\sin A}\]

\[\Rightarrow \frac{(1 - \cos A)(1 + \cos A)}{\sin^2 A}\]

\[\Rightarrow \frac{1 - \cos^2 A}{\sin^2 A}\]

By formula,

\[1 - \cos^2 A = \sin^2 A.\]

\[\Rightarrow \frac{\sin^2 A}{\sin^2 A}\]

\[\Rightarrow 1\]

Since, L.H.S. = R.H.S.