Prove the following identities I — task. Maths CBSE Live product, Class 10 (2026-27).

Prove that \(\frac{1 + \sec θ}{\sec θ} = \frac{\sin^2 θ}{1 - \cos θ}\).

Proof:

\begin{array}{l} LHS = \frac{1 + \sec θ}{\sec θ} \\ = \frac{1}{i θ} + \frac{i θ}{i θ} \\ = i θ + 1 - - - - - (1) \\ RHS = \frac{\sin^{2} θ}{1 - \cos θ} \\ = \frac{1 - i^{2} θ}{1 - \cos θ} \\ = \frac{(1 - i θ) (1 + i θ)}{1 - \cos θ} \\ = 1 + i θ - - - - - (II) \end{array}

Therefore, from \((I)\) and \((II)\), we get:

LHS \(=\) RHS

Hence proved.

Did you find an error?

2. Prove the following identities I