यदि \(\frac{{\sin \theta + \cos \theta }}{{\sin \theta - \cos \theta }} = \frac{3}{2}\) है, तो \({\sin ^4}\theta - {\cos ^4}\theta \) का मान ज्ञात कीजिए।

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Question

This question was previously asked in

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

Answer 1

दिया गया है:

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

प्रयुक्त सूत्र:

tan θ = \(\frac{sin\theta}{cos\theta}\)

तन θ = \(\frac{\ perpendicular}{\ base}\)

गणना:

2(sinθ + cosθ ) = 3(sinθ - cosθ)

2 sinθ + 2cosθ = 3 sinθ - 3 cosθ

5 cosθ = sinθ

\(\frac{sin\theta}{cos\theta}\) = 5

tanθ = 5

अब, लंब = 5 और आधार = 1

पाइथागोरस प्रमेय का प्रयोग करके,

H2 = P2 + B2

H2 = 25 + 1

H = \(\sqrt26\)

अब, sinθ = P/H और cosθ = B/H

sinθ = \(\frac{5}{\sqrt26}\)

cosθ = \(\frac{1}{\sqrt26}\)

प्रश्न के अनुसार,

\({\sin ^4}\theta - {\cos ^4}\theta \) = ( \(\frac{5}{\sqrt26}\) ) ⁴ - ( \(\frac{1}{\sqrt26}\) ) ⁴

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

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

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

∴ सही उत्तर \(\frac{12}{13}\) है।

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