TBQ - Prove the following III — task. Maths CBSE Live product, Class 10 (2025-26).

Demonstrate that:

1 + \frac{\cot^{2} θ}{1 + cosec θ} = cosec θ

Proof:

LHS \(=\)

1 + \frac{\cot^{2} θ}{1 + cosec θ}

\begin{array}{l} = 1 + \frac{(\frac{\cos^{2} θ}{i^{2} θ})}{(1 + \frac{1}{i θ})} \\ = 1 + \frac{(\frac{\cos^{2} θ}{i^{2} θ})}{(\frac{i θ + 1}{\sin θ})} \\ = 1 + \frac{i^{2} θ}{i θ (1 + \sin θ)} \\ = 1 + \frac{i^{2} θ}{\sin θ + \sin^{2} θ} \\ = \frac{i θ + i^{2} θ + \cos^{2} θ}{\sin θ + \sin^{2} θ} \\ = \frac{i θ + i}{\sin θ (1 + \sin θ)} \\ = \frac{i}{i θ} \\ = i θ \end{array}

\(=\) RHS

Thus,

1 + \frac{\cot^{2} θ}{1 + cosec θ} = cosec θ

Hence, we proved.

Did you find an error?

7. TBQ - Prove the following III