What is Manipulating Trig Identities to Solve for a Numerical Value?

[nzashadow]

Homework Statement

3(sin(x)^4+cos(x)^4)-2(sin(x)^6+cos(x)^6)=1

(these are sinx raised to the 4 and 6 powers, not x^4or6)

Homework Equations

Pythagorean Identities

The Attempt at a Solution

I've tried using pythagorean identities to change everything to terms of sine or cosine. I've been hoping to manipulate it enough to get enough cos(x)^2+sin(x)^2 to try and turn all trig functions into a numerical value. Maybe this is right and I'm missing something on the way or not going far enough. I have figured out that the 3 and the 2 are necessary to equal 1, and other values such as 2 and 1 respectively will not equate to 1, therefore (sin(x)^4+cos(x)^4)-(sin(x)^6+cos(x)^6) =/= 1-1 (although I am not sure if this is relevant.)

If anyone can give me a hint at how to correctly approach the problem that would be nice, I don't want anyone to actually work the problem out for me. Thank you

 

[Avodyne]

Try setting $y=(\sin x)^2$, and writing everything in terms of $y$.

 

[Apphysicist]

I rewrote it as 3((sin²)²+cos⁴)-2(sin²*sin⁴+cos⁶)

And then using sin²=1-cos², and some FOILing, the messy algebra worked out nicely.

 

[nzashadow]

Thanks guys, got it.