Prove that: $\left(\frac{1}{\cos heta} - \cos heta ight)\left(\frac{1}{\sin heta} - \sin heta ight) = \frac{1}{ an heta + \cot heta}$

Question

Prove that: $\left(\frac{1}{\cos	heta} - \cos	hetaight)\left(\frac{1}{\sin	heta} - \sin	hetaight) = \frac{1}{	an	heta + \cot	heta}$

Accepted Answer

Step 1: LHS $= \frac{1 - \cos^2	heta}{\cos	heta} 	imes \frac{1 - \sin^2	heta}{\sin	heta}$
Step 2: $= \frac{\sin^2	heta}{\cos	heta} 	imes \frac{\cos^2	heta}{\sin	heta} = \sin	heta\cos	heta$
Step 3: RHS $= \frac{1}{	an	heta + \cot	heta} = \frac{1}{\frac{\sin	heta}{\cos	heta} + \frac{\cos	heta}{\sin	heta}}$
Step 4: $= \frac{1}{\frac{\sin^2	heta + \cos^2	heta}{\cos	heta\sin	heta}} = \sin	heta\cos	heta$. $	herefore$ LHS $=$ RHS

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