Q) Prove that (cosec A – sin A) (sec A- cos A) = \frac{1}{(\cot A + \tan A)}

Ans:  Let’s start with LHS:

(cosec A – sin A) (sec A- cos A)

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

=   (\frac{1 - \sin^2 A}{\sin A}) (\frac{1 - \cos^2 A}{\cos A})

=   (\frac{\cos^2 A}{\sin A}) (\frac{\sin^2 A}{\cos A})

=   cos A . sin A

Now, Let’s solve RHS:

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

=      \frac{1}{\frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}}

=     \frac{\sin A \cos A}{\cos^2 A + \sin^2 A}

= sin A . cos A ………… same as LHS…. Hence Proved!

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