Prove that: (1/cosθ – cos θ) (1/(sin θ) – sin θ) = 1/(tan θ+ cot θ)

Question

Q) Prove that: $(\frac{1}{\cos	heta} – \cos 	heta)$ x $(\frac{1}{\sin	heta} – \sin 	heta) =  \frac{1}{(	an	heta + \cot 	heta)}$

Sapling Academy · Accepted Answer

Let’s start from LHS

LHS = $(\frac{1}{\cos	heta} – \cos 	heta)$ x $(\frac{1}{\sin	heta} – \sin 	heta)$

= $(\frac{1 - \cos^2	heta}{\cos 	heta})$ x $(\frac{1 - \sin^2	heta}{\sin	heta})$

= $(\frac{\sin^2	heta}{\cos	heta})$  x $(\frac{\cos^2	heta}{\sin	heta})$

= sin θ x cos θ

Now, take RHS

RHS = $\frac{1}{(	an	heta + \cot 	heta)}$

= $\frac{1}{\frac{\sin	heta|}{\cos	heta} + \frac{\cos 	heta}{\sin	heta}}$

= $\frac{1}{\frac{\sin ^2 	heta| + \cos ^2	heta}{\sin 	heta \cos	heta}}$

= sin θ x cos θ

Since LHS = RHS

Hence proved!

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