If sin θ sec² θ = 2/3, 0⁰ θ 90⁰ then the value of (tan² θ + sin² θ) is:

If sin θ sec² θ = 2/3, 0⁰ θ 90⁰ then the value of (tan² θ…

If sin $\theta$ sec² $\theta$ = 2/3, 0⁰ < $\theta$ < $90^0$ then the value of (tan² θ + sin² θ) is:

Show answer & explanation

Correct answer: A

Explanation

The correct answer is Option 1 i.e. 7/12.

sin θ × sec² θ = 2/3
sec²θ = 1 + tan² θ = 1 + x
So, sin θ = 2 / [3(1 + x)]
sin² θ = x / (1 + x)
sin² θ = 4 / [9(1 + x)²]

x / (1 + x) = 4 / [9(1 + x)²]
⇒ x(1 + x) = 4/9
⇒ x² + x = 4/9
⇒ 9x² + 9x − 4 = 0
⇒ x = 1/3

tan² θ + sin² θ = x + x / (1 + x)
= 1/3 + (1/3) / (4/3) = 1/3 + 1/4 = 7/12