Question

If $ \sin\theta + \csc\theta = 4 $, then find: $$ \sin^2\theta + \csc^2\theta = ? $$

• A. 12
• B. 18
• C. 16
• D. 14

Hint:

Use identities like \( a + \frac{1}{a} = k \Rightarrow a^2 + \frac{1}{a^2} = k^2 - 2 \).

Answer 1

The Correct Option is D

We are given: \[ \sin\theta + \csc\theta = 4 \] Let \( x = \sin\theta \), then \( \csc\theta = \frac{1}{x} \) So: \[ x + \frac{1}{x} = 4 \Rightarrow x^2 + \frac{1}{x^2} = ? \] We square both sides: \[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 = 16 \Rightarrow x^2 + \frac{1}{x^2} = 16 - 2 = \boxed{14} \] Which is: \[ \sin^2\theta + \csc^2\theta = \boxed{14} \]