Question

How do you simplify the expression `(cos^3t+sin^3t)/(cost+sint)^2`?

Answer 1

`(2 - sin 2t)/(2(cos t + sin t))`

Explanation:

Expression: `E = (cos^3 t + sin^3 t)/(cos t + sin t)^2`
Use these algebraic identities:
`a^3 + b^3 = (a + b)(a^2 - ab + b^2)`
`cos^2 t + sin^2 t = 1`
`sin 2t = 2sin t.cos t`.
We get:
`cos^3 t + sin^3 t = (cos t + sin t)( cos^2 t + sin^2 t - sin t.cos t) =`
`= (cos t + sin t)( 1 - (sin 2t)/2) = (cos t + sin t)(1/2)(2 + sin 2t)`
Simplify by (cos t + sin t), we get
`E = (2 - sin 2t)/(2(cos t + sin t))`