जर \(\frac{{\sin \theta + \cos \theta }}{{\sin \theta - \cos \theta }} = \frac{3}{2}\), तर \({\sin ^4}\theta - {\cos ^4}\theta \) याचे मूल्य असेल :

Question

Question

This question was previously asked in

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

Answer 1

दिलेली माहिती:

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

वापरलेले सूत्र:

tanθ = \(\frac{sinθ}{cosθ}\)

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

गणना:

2(sinθ + cosθ ) = 3(sinθ - cosθ)

2 sinθ + 2cosθ = 3 sinθ - 3 cosθ

5 cosθ = sinθ

\(\frac{sinθ}{cosθ}\) = 5

tanθ = 5

आता, लंब= 5 आणि पाया = 1

पायथागोरसचे प्रमेय वापरुन,

H2 = P2 + B2

H2 = 25 + 1

H = \(√26\)

आता, sinθ = P/H आणि cosθ = B/H

sinθ = \(\frac{5}{√26}\)

cosθ = \(\frac{1}{√26}\)

प्रश्नानुसार,

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

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

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

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

∴ \(\frac{12}{13}\) हे योग्य उत्तर आहे.