Question

Evaluate `sin^4 15^@ + cos^4 15^@`?

Answer 1

The expression gives a result of `7/8`

Explanation:

We can see that

`(sin^2x + cos^2x)^2 = (sin^2x + cos^2x)(sin^2x+ cos^2x) = sin^4x + cos^4x + 2cos^2xsin^2x`

Therefore,

`sin^4x + cos^4x = (sin^2x + cos^2x)^2 - 2cos^2xsin^2x`

We seek to find:

`(sin^2(15˚) + cos^2(15˚))^2 - 2cos^2(15˚)sin^2(15˚) = sin^4(15˚) + cos^4(15˚)`

We know that `sin^2x+ cos^2x = 1`, therefore,

`1^2 - 2cos^2(15˚)sin^2(15˚) = sin^4(15˚) + cos^4(15˚)`

Now we notice that `sin^2(2x) = 4sin^2xcos^2x`, therefore,

`1^2 - 1/2sin^2(2(15˚)) = sin^4(15˚) + cos^4(15˚)`

`1^2 - 1/2sin^2(30˚) = sin^4(15˚) + cos^4(15˚)`

`1 - 1/2(1/2)^2 = sin^4(15˚) + cos^4(15˚)`

`7/8 = sin^4(15˚) + cos^4(15˚)`

And if we check by calculator, we see that we do indeed get `7/8`.

Hopefully this helps!