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

Q) Prove that: $(\frac{1}{\cos\theta} - \cos \theta)$ x $(\frac{1}{\sin\theta} - \sin \theta) = \frac{1}{(\tan\theta + \cot \theta)}$

Ans:

Let’s start from LHS

LHS = $(\frac{1}{\cos\theta} - \cos \theta)$ x $(\frac{1}{\sin\theta} - \sin \theta)$

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

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

= sin θ x cos θ

Now, take RHS

RHS = $\frac{1}{(\tan\theta + \cot \theta)}$

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

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

= sin θ x cos θ

Since LHS = RHS

Hence proved!

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