Question
Download Solution PDFWhat is 3 (sin x - cos x)4 + 6 (sin x + cos x)2 + 4 (sin x)6 + 4 (cos x)6 equal to ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFPut x = 0,
3 (sin x - cos x)4 + 6 (sin x + cos x)2 + 4 (sin x)6 + 4 (cos x)6
⇒ 3 (sin 0 - cos 0)4 + 6 (sin 0 + cos 0)2 + 4 (sin 0)6 + 4 (cos 0)6
⇒ 3 (-1)4 + 6 (1)2 + 4 (0)6 + 4 (1)6
⇒ 3 + 6 + 4 = 13
∴ 3 (sin x - cos x)4 + 6 (sin x + cos x)2 + 4 (sin x)6 + 4 (cos x)6 equal to 13.
Alternate MethodConcept Used:
(a + b)2 = a2 + b2 + 2ab
(a - b)2 = a2 + b2 - 2ab
sin2 θ + cos2 θ = 1
Calculation:
⇒ 3 [sin x - cos x]4 + 6 [sin x + cos x]2 + 4 [sin6 x + cos6 x]
⇒ 3 {(sin x - cos x)2}2 + 6 (sin2 x + cos2 x + 2 sin x cos x) + 4 [(sin2 x)3 + (cos2 x)3]
⇒ 3 {(sin2 x + cos2 x - 2 sin x cos x)2} + 6 (1 + 2 sin x cos x) + 4 [(sin2 x + cos2 x)(sin4 x + cos4 x - sin2 x cos2 x)]
⇒ 3 (1 - 2 sin x cos x)2 + 6 (1 + 2sin x cos x) + 4 [(sin2 x)2 + (cos2 x)2 + 2 sin2 x cos2 x - 3 sin2 x cos2 x]
⇒ 3 (1 + 4 sin2 x cos2 x - 4sin x cos x) + 6 (1 + 2sin x cos x) + 4 [(sin2 x + cos2 x)2 - 3 sin2 x cos2 x]
⇒ 3 + 12 sin2 x cos2 x - 12 sin x cos x + 6 + 12 sin x cos x + 4 - 12 sin2 x cos2 x
⇒ 13
∴ 3 (sin x - cos x)4 + 6 (sin x + cos x)2 + 4 (sin x)6 + 4 (cos x)6 equal to 13.
Last updated on May 29, 2025
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