Question

If cos θ + sec θ = k, then what is the value of sin²θ - tan²θ ?
1. 4 - k
2. 4 - k²
3. k² - 4
4. k² + 2

Answer 1

Correct Answer - Option 2 : 4 - k²

Given:

cos θ + sec θ = k

Formula used:

sec θ = 1/cos θ 

a² + b² = (a + b)² - 2ab

Calculation:

cos θ + sec θ = k

⇒ cos θ + 1/cos θ = k

On squaring both sides, we get,

⇒ cos²θ + 1/cos²θ + 2 = k²

⇒ cos2θ + 1/cos2θ = k² - 2

According to the question,

sin2θ - tan2θ = (1 - cos²θ) - (sec²θ - 1)

⇒ 2 - (cos2θ + 1/cos2θ)

⇒ 2 - (k2 - 2)

⇒ sin2θ - tan2θ = 4 - k²

∴ The value of sin2θ - tan2θ is 4 - k².