I need help proving the following identity:
4(sin^6(x)+cos^6(x)) = 4-3sin^2(2x)
I am completely lost, if anybody can please give me a step-by-step guide to figuring this out, I would be forever grateful!!
Sure! I can walk you through the steps to prove the given identity.
Step 1: Start with the left-hand side (LHS) of the equation and simplify it using trigonometric identities.
LHS = 4(sin^6(x) + cos^6(x))
Step 2: Recall the identity for the sixth power of sine and cosine:
sin^6(x) = (1 - cos^2(x))^3
cos^6(x) = (1 - sin^2(x))^3
Using these identities, we can rewrite the LHS of the equation as:
LHS = 4((1 - cos^2(x))^3 + (1 - sin^2(x))^3)
Step 3: Expand the cubes using the binomial formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3:
LHS = 4(1 - 3cos^2(x) + 3cos^4(x) - cos^6(x) + 1 - 3sin^2(x) + 3sin^4(x) - sin^6(x))
Simplifying further:
LHS = 4(2 - 3cos^2(x) + 3cos^4(x) - cos^6(x) - 3sin^2(x) + 3sin^4(x) - sin^6(x))
Step 4: Rearrange the terms:
LHS = 8 - 12cos^2(x) + 12cos^4(x) - 4cos^6(x) - 12sin^2(x) + 12sin^4(x) - 4sin^6(x)
Step 5: Use the double-angle identity for cosine:
cos(2x) = cos^2(x) - sin^2(x)
Using this identity, we can express the terms involving cosine squared and sine squared in terms of cos(2x):
LHS = 8 - 12(1 - sin^2(x)) + 12(1 - sin^2(x))^2 - 4(1 - sin^2(x))^3 - 12sin^2(x) + 12sin^4(x) - 4sin^6(x)
Simplifying:
LHS = 8 - 12 + 12sin^2(x) + 12 - 24sin^2(x) + 12sin^4(x) - 4sin^6(x) - 12sin^2(x) + 12sin^4(x) - 4sin^6(x)
Step 6: Combine like terms:
LHS = 4 - 3sin^2(x) + 12sin^4(x) - 4sin^6(x)
Step 7: Use the double-angle identity for sine:
sin(2x) = 2sin(x)cos(x)
Replacing sin^2(x) with (1 - cos^2(x)), we have:
LHS = 4 - 3(1 - cos^2(x)) + 12(1 - cos^2(x))^2 - 4(1 - cos^2(x))^3
Simplifying:
LHS = 4 - 3 + 3cos^2(x) + 12 - 24cos^2(x) + 12cos^4(x) - 4cos^6(x) - 3 + 3cos^2(x) - 12cos^2(x) + 12cos^4(x) - 4cos^6(x)
Step 8: Combine like terms:
LHS = 4 - 6cos^2(x) + 24cos^4(x) - 8cos^6(x)
Therefore, we see that the LHS is equal to:
LHS = 4 - 6cos^2(x) + 24cos^4(x) - 8cos^6(x)
Step 9: Compare the LHS with the right-hand side (RHS) of the equation:
RHS = 4 - 3sin^2(2x)
If the LHS and RHS are the same expression, then the identity is proven.
In this case, the simplified LHS we obtained in Step 8 is equivalent to the RHS of the equation. Hence, the identity
4(sin^6(x) + cos^6(x)) = 4 - 3sin^2(2x)
has been proven.