If tan²θ + cot²θ = 50, then find sec⁴θ + csc⁴θ.
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Correct answer: A
Explanation
tan²θ + cot²θ = 50
(tan²θ + cot²θ) = (tan²θ + cot²θ + 2) − 2 = (tanθ + cotθ)² − 2
sec²θ = 1 + tan²θ, csc²θ = 1 + cot²θ
tan²θ + cot²θ + 2 = 52
⇒ (tanθ + cotθ)² = 52
sec⁴θ + csc⁴θ
= (1 + tan²θ)² + (1 + cot²θ)²
= 1 + 2tan²θ + tan⁴θ + 1 + 2cot²θ + cot⁴θ
= 2 + 2(tan²θ + cot²θ) + (tan⁴θ + cot⁴θ)
tan⁴θ + cot⁴θ = (tan²θ + cot²θ)² − 2tan²θ cot²θ= 50² − 2= 2498
Now,
sec⁴θ + csc⁴θ = 2 + 2×50 + 2498= 2 + 100 + 2498= 2600