If \(\frac{{\sin \theta + \cos \theta }}{{\sin \theta - \cos \theta }} = \frac{3}{2}\), then the value of \({\sin ^4}\theta - {\cos ^4}\theta \) is:

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 09 Dec 2022 Shift 2)

Answer 1

Given:

\(\frac{{\sin θ + \cos θ }}{{\sin θ - \cos θ }} = \frac{3}{2}\)

Formula used:

tanθ = \(\frac{sinθ}{cosθ}\)

tanθ = \(\frac{\ perpendicular}{\ base}\)

Calculation:

2(sinθ + cosθ ) = 3(sinθ - cosθ)

2 sinθ + 2cosθ = 3 sinθ - 3 cosθ

5 cosθ = sinθ

\(\frac{sinθ}{cosθ}\) = 5

tanθ = 5

now, perpendicular = 5 and base = 1

Using Pythagoras's theorem,

H² = P² + B²

H² = 25 + 1

H = \(√26\)

Now, sinθ = P/H and cosθ = B/H

sinθ = \(\frac{5}{√26}\)

cosθ = \(\frac{1}{√26}\)

According to the question,

\({\sin ^4}θ - {\cos ^4}θ \) = (\(\frac{5}{√26}\))⁴ - (\(\frac{1}{√26}\))⁴

\(=\frac{625}{676}\)-\(\frac{1}{676}\)

=\(\frac{625-1}{676}\)

=\(\frac{624}{676}\\=\frac{12}{13}\)

∴ The correct answer is \(\frac{12}{13}\).

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