Question

Given that
`sin(x)+cos(x)=(2)/(5)`, compute the following.


`sin^3(x)+cos^3(x)`

The question also says to use the formula of the sum of two cubes and the formula of perfect square trinomial

Also the answer is no 92/125 i already tried that

Answer 1

Answer: 71/125

Explanation:

`\sin^3 x+\cos^3 x=(\sin x+\cos x)(\sin^2 x+\cos^2 x-\sin x \cos x)\\\\=\left((2)/(5) \right)(1-\sin x \cos x)`

To find
`\sin x \cos x`, we can notice that
`(\sin x +\cos x)^2 =\sin^2 x+\cos ^2 x+\boxed{2\sin x \cos x}`

Applying this, we get that:

`(2/5)^2 =1+2\sin x \cos x\\\\-(21)/(25)=2\sin x \cos x\\\\\sin x \cos x=-(21)/(50)`

`\therefore \sin^3 x+\cos^3 x=(2/5)(71/50)=(71)/(125)`