Question

x = (cosec A + cot A) (1 − cos A).

Assertion (A): x = sin A.

Reason (R): \[x = \left(\frac{1}{\sin A} + \frac{\cos A}{\sin A}\right)(1 - \cos A)\]

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

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Given, \[x = (\operatorname{cosec} A + \cot A)(1 - \cos A)\]

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

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

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

\[{} = \left(\frac{1 - \cos^2 A}{\sin A}\right)\]

\[{} = \left(\frac{\sin^2 A}{\sin A}\right)\]

\[{} = \sin A.\]

∴ Both A and R are true and R is correct reason for A.