write the value of 2(sin^6+cos^6)-3(sin^4+cos^4)+1
To find the value of the given expression, let's break it down step by step.
We have the expression: 2(sin^6 + cos^6) - 3(sin^4 + cos^4) + 1
Before simplifying further, let's write sin^6 and cos^6 in terms of sin^2 and cos^2.
Using the identity sin^2(x) = 1 - cos^2(x) and cos^2(x) = 1 - sin^2(x), we can rewrite the expression as:
2[(1 - cos^2(x))^3 + cos^6(x)] - 3[(1 - sin^2(x))^2 + sin^4(x)] + 1
Next, let's expand the higher powers of (1 - cos^2(x)) and (1 - sin^2(x)):
2[(1 - 3cos^2(x) + 3cos^4(x) - cos^6(x)) + cos^6(x)] - 3[(1 - 2sin^2(x) + sin^4(x)) + sin^4(x)] + 1
Simplifying further, we have:
2[1 - 3cos^2(x) + 3cos^4(x)] - 3[1 - 2sin^2(x) + sin^4(x)] + 1
Now, distribute the coefficients:
2 - 6cos^2(x) + 6cos^4(x) - 3 + 6sin^2(x) - 3sin^4(x) + 1
Combine like terms:
6cos^4(x) - 6cos^2(x) - 3sin^4(x) + 6sin^2(x) - 2
And that is the simplified value of the expression 2(sin^6 + cos^6) - 3(sin^4 + cos^4) + 1.