Question
In: Math
If (sin4x)/2 + (cos4x)/3 = ⅕, then
If (sin4x)/2 + (cos4x)/3 = ⅕, then
(a) tan2x = ⅔
(b) (sin8x)/8 + (cos8x)/27 = 1/125
(c) tan2x = ⅓
(d) (sin8x)/8 + (cos8x)/27 = 2/125
Solutions
Expert Solution
Given (sin⁴x)/2 + (cos⁴x)/3 = ⅕
=> (sin⁴x)/2 + (1 – sin²x)2/3 = ⅕
=> (sin⁴x)/2 + (1 – 2 sin²x + sin⁴x)/3 = 1/5
Put t = sin²x
=> t²/2 + (1-2t+t²)/3 = ⅕
=> 3t² + 2 – 4t + 2t² = 6/5
=> 5t² – 4t + 2 = 6/5
=> 25t² – 20t + 10 = 0
=> (5t-2)² = 0
=> t = 2/5
sin²x = 2/5
cos²x = 1-sin²x
= 1 – 2/5
= ⅗
tan²x = 2/3
(sin⁸x)/8 + (cos⁸x)/27 = (2/5)4/8 + (3/5)4/27
= (2/54) + (3/54)
= 5/54
= 1/125
Hence option a and b are correct.
tan²x = ⅔ and (sin⁸x)/8 + (cos⁸x)/27 = 1/125 are correct.
Nikhil Thakur answered 2 years agoRelated Solutions
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