Question

\[\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta} = x.\]

Statement (1): x = 1.

Statement (2): \[x = \frac{\sin^2 \theta - 1}{\cos^2 \theta} = \frac{-\cos^2 \theta}{\cos^2 \theta} = -1\]

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

Statement 1 is false, and statement 2 is true.

Explanation:

Given, \[\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta} = x\]

\[\Rightarrow x = \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta}\]

\[\Rightarrow x = \frac{\sin^2 \theta - 1}{\cos^2 \theta}\]

\[\Rightarrow x = \frac{\sin^2 \theta - (\sin^2 \theta + \cos^2 \theta)}{\cos^2 \theta}\]

\[\Rightarrow x = \frac{\sin^2 \theta - \sin^2 \theta - \cos^2 \theta}{\cos^2 \theta}\]

\[\Rightarrow x = \frac{-\cos^2 \theta}{\cos^2 \theta}\]

\[\Rightarrow x = -1\]

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