Q.413. Trigonometry proof

Question 413.
Given : sin^4 θ/a  + cos^4  θ/b = 1/(a + b).
To Prove : sin^8  θ/a^3 + cos^8  θ/b^3 = 1/(a + b)^3.

Answer 413.
(1/a) sin^4 θ + (1/b) cos^4 θ = 1/(a + b) => (1/a) sin^4 θ + (1/b) (1 - sin^2 θ)^2 = 1/(a +b)

=> b(a + b) sin^4 θ + a (a + b) (1 - sin^2 θ)^2 = ab
=> (a + b)^2 sin^4 θ - 2a (a + b) sin^2 θ + a (a + b) = ab
=> (a + b)^2 sin^4 θ - 2a (a + b) sin^2 θ + a^2 = 0
=> [(a + b) sin^2 θ - a]^2 = 0
=> sin^2 θ = a/(a + b)
and cos^2 θ = 1 - sin^2 θ = 1 - a/(a + b) = b/(a + b)

=> (1/a^3) sin^8 θ + (1/b^3) cos^8 θ
= (1/a^3) * a^4/(a + b)^4 + (1/b^3) * b^4 / (a + b)^4
= (a + b) / (a + b)^4
= 1/(a + b)^3. (proved).

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