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Questions
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Prove the following:
`(sin^3 θ + cos^3 θ)/(sin θ +cos θ) + (sin^3 θ - cos ^3 θ)/(sin θ - cos θ) = 2`
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Solution
LHS= `(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) `
=` ((cos θ + sin θ)(cos^2 θ - cos θ sin θ + sin^2 θ))/((cos θ + sin θ)) + ((cos θ - sin θ)(cos^2 θ + cos θ sin θ + sin^2 θ))/((cos θ - sin θ))`
= (cos2 θ + sin2 θ − cos θ sin θ) + (cos2 θ + sin2 θ + cos θ sin θ)`
= (1 − cos θ sin θ) + (1 + cos θ sin θ)
= 2
= RHS
Hence, LHS = RHS
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