If sin θ + tan θ = p and tan θ - sin θ = q, then which of the relations satisfies for given values of p and q?

If sin θ + tan θ = p and tan θ - sin θ = q, then which of…

If sin $\theta$ + tan $\theta$ = p and tan $\theta$ - sin $\theta$ = q, then which of the relations satisfies for given values of p and q?

Show answer & explanation

Correct answer: A

Explanation

The correct answer is option 1 i.e. $p^2-q^2=4\sqrt{pq}$ .

sin θ + tan θ = p

⇒ tan θ + sin θ = p .....(1)

tan θ - sin θ = q .....(2)

So, pq = (sin θ + tan θ)(tan θ - sin θ) = tan²θ - sin²θ

= sin²θ(1/cos²θ - 1)

= sin²θ(sec²θ - 1)

= sin²θ tan²θ

⇒ √(pq) = sinθ tanθ

p² - q² = (tan θ + sin θ)² - (tan θ - sin θ)² = 4sinθ tanθ = 4√(pq)

Hence, $p^2-q^2=4\sqrt{pq}$