Question

If tan⁴θ – cot²θ = 1, and θ < 90° then find the value of sin²θ + sin³θ 

1. 1
2. 2
3. 3
4. 4

Answer 1

Correct Answer - Option 1 : 1

Given:

tan⁴θ – cot²θ = 1

Formulae used:

1) tanθ = sinθ/cosθ

2) secθ = 1/cosθ

3) 1 + cot²θ = cosec²θ

4) sin²θ + cos²θ = 1

Calculations:

tan⁴θ – cot²θ = 1

⇒ tan⁴θ = 1 + cot²θ

⇒ tan⁴θ = cosec²θ

⇒ tan²θ = cosecθ

⇒ sin²θ/cos²θ = 1/sinθ

⇒ sin³θ = cos²θ

 Now,

sin2θ + sin3θ = cos2θ + sin2θ

⇒ cos²θ + sin²θ

⇒ 1

∴ The value of sin2θ + sin3θ is 1