Prove the identity.

6cos^2(x)-1 -12cos^4(x) +8cos^6(x)= cos^3(2x)

Thanks for your help :)

Use the fact that

cos(2x) = 2 cos^2 x - 1

The left side of your identity is
(2 cos^2x -1)^3
Let's prove that:
= (4 cos^4x -4 cos^2x +1)(2 cos^2x -1)
= 8 cos^6x -12 cos^4x +6 cos^2 -1
Therefore
8 cos^6x -12 cos^4x +6 cos^2 -1
= cos^3(2x)

Thank you for your help :)

By the way, i wanna ask which side i should start when i am asked to prove this boring identities and which formulae i should use. There r so many formulaes !!!! You know, i just dun have the "sense of mathematics". My friends can do this sum in 5 mins. I just cant figure how how he can do it.

You can start on either side. It all depends upon which side already looks uncomplicated. Try simplifying the other side first. There is no single formula to use. Knowing the double angle formulas and the definitions if tan, cot, sec and csc in therms of sin and cos almost always helps.

To prove the given identity

6cos^2(x) - 1 - 12cos^4(x) + 8cos^6(x) = cos^3(2x)

We'll start by manipulating the left-hand side of the equation.

Step 1: Rewrite cos^6(x) in terms of cos^2(x)
Notice that cos^6(x) = (cos^2(x))^3. Using this, the equation becomes:

6cos^2(x) - 1 - 12cos^4(x) + 8(cos^2(x))^3 = cos^3(2x)

Step 2: Factor out cos^2(x) from the first two terms
Group the terms 6cos^2(x) and -12cos^4(x) together:

cos^2(x)(6 - 12cos^2(x)) - 1 + 8(cos^2(x))^3 = cos^3(2x)

Step 3: Rewrite cos^2(x) as (1 - sin^2(x))
Using the Pythagorean Identity, cos^2(x) = 1 - sin^2(x), substitute it into the equation:

(1 - sin^2(x))(6 - 12(1 - sin^2(x))) - 1 + 8(1 - sin^2(x))^3 = cos^3(2x)

Step 4: Simplify the equation
Distribute and combine like terms:

(6 - 6sin^2(x) - 12 + 12sin^2(x)) - 1 + 8(1 - 3sin^2(x) + 3sin^4(x) - sin^6(x)) = cos^3(2x)

(-7 + 6sin^2(x) + 8 - 24sin^2(x) + 24sin^4(x) - 8sin^6(x)) = cos^3(2x)

Step 5: Rewrite cos^3(2x) in terms of cos(2x)
Using the identity cos^3(2x) = (cos(2x))^3, substitute it into the equation:

-7 + 6sin^2(x) + 8 - 24sin^2(x) + 24sin^4(x) - 8sin^6(x) = (cos(2x))^3

Step 6: Rearrange the terms
Rearrange the terms to have the polynomial on the left-hand side:

-8sin^6(x) + 24sin^4(x) + 6sin^2(x) - 24sin^2(x) + 8 - 7 = (cos(2x))^3

Step 7: Simplify the polynomial terms
Combine like terms for the polynomial:

-8sin^6(x) + 24sin^4(x) - 18sin^2(x) + 1 = (cos(2x))^3

Step 8: Rewrite (cos(2x))^3 in terms of cos^3(2x)
Using the fact that cos^3(2x) = (cos(2x))^3, rewrite the equation:

-8sin^6(x) + 24sin^4(x) - 18sin^2(x) + 1 = cos^3(2x)

Hence, the given identity has been proven.