Question

Let y = sin²θ + cos²θ + tan²θ + sec²θ + cosec²θ + cot²θ attains its least value (where θ ∈ [0, 4π]), then number of such possible values of θ is ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

Let y = sin²θ + cos²θ + tan²θ + sec²θ + cosec²θ + cot²θ attains its least value (where θ ∈ [0, 4π]), then number of such possible values of θ is 8.00.

Explanation:

y = 1 + (tan²θ + cot²θ) + (sec²θ + cosec²θ)

y = 1 + sec²θ – 1 + cosec²θ – 1 + sec²θ + cosec²θ

⇒ y = 2(sec2θ + cosec2θ) – 1

= `2/(sin^2θcos^2θ) - 1`

⇒ y = 8cosec²θ = 1

y is least if cosec²2θ = 1

∴ y_least = 8 – 1 = 7

cosec2θ = ±1, 2θ ∈ (0, 8π)

2θ = `π/2, (3π)/2, (5π)/2, (7π)/2, (9π)/2, (11π)/2, (13π)/2, (15π)/2`

Total = 8