Question

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\text{(cosec A - sin A)(sec A - cos A)} =\frac{ 1}{\text{(tan A + cot A)}}\) [Hint : Simplify LHS and RHS separately]

Answer 1

\(\text{(cosec A - sin A)(sec A - cos A)} = \frac{1}{\text{(tan A + cot A)}}\)

L.H.S =\(\text{ (cosec A - sin A)(sec A - cos A)}\)

Substitute cosec A = 1/sin A and sec A = 1/cos A:

\(\Rightarrow (\frac{1}{\text{sin A }}-\text{ sin A})(\frac{1}{\text{cos A }}- \text{cos A})\)

Combine terms within each parenthesis:

\(= \frac{\text{(1 - sin² A)}}{\text{sin A }}×\frac{\text{ (1 - cos² A)}}{\text{cos A}}\)

Apply the identity sin² A + cos² A = 1, so 1 - sin² A = cos² A and 1 - cos² A = sin² A:

\(= \frac{\text{cos² A}}{\text{sin A }}×\frac{\text{ sin² A}}{\text{cos A}}\)

Simplify the expression:

\(= \frac{\text{cos A sin A}}{1}\)

Rewrite the denominator using the identity sin² A + cos² A = 1:

\(=\frac{\text{ sin A cos A}}{\text{(sin² A + cos² A)}}\)

Divide the numerator and denominator by sin A cos A:

\(= \frac{1}{[(\frac{\text{sin² A}}{\text{sin A cos A}}) + (\frac{\text{cos² A}}{\text{sin A cos A}})]}\)

Simplify the terms in the denominator:

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

Substitute tan A = sin A / cos A and cot A = cos A / sin A:

\(= \frac{1}{\text{(tan A + cot A)}}\)

= RHS