I've tried many times to get the equation to balance using trig identities like half-angle formulas, power-reducing formulas, and double angle formulas, but I couldn't get the equation to equal out. I only need to work one side to make it equal to the other, like a kind of proof. Please help!
1 - 4sin^2(x)cos^2(x) = cos^2(2x)
Can you check your book to see if the given expression is "supposed" to be an identity, or is it a question to solve for x?
The left hand side:
4sin²(x)cos²(x)
=(2sin(x)cos(x))²
=sin²(2x)
Which is not an identity with the right hand side.
If it is an equation, try to solve for x. It would be an excellent exercise.
To help you balance the equation 1 - 4sin^2(x)cos^2(x) = cos^2(2x), we can start by simplifying the left side using trigonometric identities.
1. Start with the left side of the equation:
1 - 4sin^2(x)cos^2(x)
2. Let's use the double angle formula for cosine: cos(2θ) = cos^2(θ) - sin^2(θ).
Replace cos^2(2x) with cos^2(x) - sin^2(x).
The equation becomes: 1 - 4sin^2(x)cos^2(x) = cos^2(x) - sin^2(x)
3. Now, let's use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Replace sin^2(x) with 1 - cos^2(x) and simplify:
1 - 4(1 - cos^2(x))cos^2(x) = cos^2(x) - (1 - cos^2(x))
4. Expand and simplify:
1 - 4cos^2(x) + 4cos^4(x) = cos^2(x) - 1 + cos^2(x)
5. Rearrange terms:
4cos^4(x) - 4cos^2(x) + 2 = 0
Now, you have a quadratic equation in terms of cos(x). You can solve this equation by factoring or using the quadratic formula to find the values of cos(x) that satisfy the equation.
Once you find the values of cos(x), you can substitute them back into the original equation to check if they balance the equation.