8 sin²(3x) cos²(3x) = 2 [2 sin(3x) cos(3x)]²
= 2 [sin(6x)]²
= 2[1 - cos(12x)]/2
= 1 - cos(12x)
good job
I usually pick some angle, say x = 21,
and sub it into the original, and the final.
of course they have to be the same.
This does of course not "prove" your answer is right, but the probability that you would get the same result if you had an error would be "very small"
Of course if you don't get the same result, then you know you are wrong.
To solve the equation 8 sin²(3x) cos²(3x) = 2 [2 sin(3x) cos(3x)]², we can simplify step by step:
Step 1: Start with the left-hand side (LHS) of the equation and apply trigonometric identities.
8 sin²(3x) cos²(3x)
= 2 [2 sin(3x) cos(3x)]²
Here, we recognize the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ). Therefore, we can rewrite the RHS of the equation as sin²(6x).
Step 2: Substitute sin²(6x) for [2 sin(3x) cos(3x)]².
8 sin²(3x) cos²(3x) = 2 [sin(6x)]²
Step 3: Simplify the expression inside the brackets.
2 [sin(6x)]² = 2(1 - cos²(6x))
Using the identity sin²(θ) = 1 - cos²(θ)
Step 4: Further simplify the expression.
2(1 - cos²(6x)) = 2(1 - cos(12x))/2
Using the double-angle identity for cosine: cos(2θ) = 1 - 2sin²(θ)
Step 5: Divide both sides of the equation by 2.
2(1 - cos(12x))/2 = 1 - cos(12x)
Finally, we have simplified the original equation to 8 sin²(3x) cos²(3x) = 2 [2 sin(3x) cos(3x)]² = 2 [sin(6x)]² = 2(1 - cos(12x))/2 = 1 - cos(12x).