Question

\[\cos^2 \theta + \frac{1}{1 + \cot^2 \theta} = x.\]

Statement (1): x = 1.

Statement (2): \[x = \cos^2 \theta + \frac{1}{\operatorname{cosec}^2 \theta}\]

\[{} = \cos^2 \theta + \sin^2 \theta\]

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Given, \[\cos^2 \theta + \frac{1}{1 + \cot^2 \theta} = x\]

\[\Rightarrow x = \cos^2 \theta + \frac{1}{1 + \cot^2 \theta}\]

\[\Rightarrow x = \cos^2 \theta + \frac{1}{\operatorname{cosec}^2 \theta}\]

\[\Rightarrow x = \cos^2 \theta + \sin^2 \theta\]

\[\Rightarrow x = 1\]

∴ Both the statements are true.