Prove that : (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) = 1

Q) Prove that : (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) = 1

Ans: Let’s start with LHS:

LHS: (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ)

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

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

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

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

= $\cancel{(cos \theta sin \theta)} (\frac{1}{\cancel{sin\theta cos\theta}})$

= 1

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