2cos^2a*cos^2b+2sin^2asin^2b-1=cos2acos2b
To prove the equation 2cos^2a*cos^2b + 2sin^2asin^2b - 1 = cos2acos2b, we'll simplify both sides of the equation and make use of trigonometric identities.
Starting with the left-hand side (LHS) of the equation:
LHS = 2cos^2a*cos^2b + 2sin^2asin^2b - 1
To simplify this expression, we'll make use of the following trigonometric identities:
1. cos^2x + sin^2x = 1 (Pythagorean identity)
2. cos2x = 2cos^2x - 1
First, let's rewrite the expression using the Pythagorean identity:
LHS = 2cos^2a * cos^2b + 2sin^2a * sin^2b - 1
Now let's substitute cos2a and cos2b using identity (2):
LHS = cos2a/2 * cos^2b + cos2b/2 * sin^2a - 1
Next, using the product-to-sum identities, let's express cos2a/2 * cos^2b and cos2b/2 * sin^2a as sums:
LHS = (cos2a/2) * (cos^2b) + (cos2b/2) * (sin^2a) - 1
= (1/2) * (cos2a * cos^2b + cos2b * sin^2a) - 1
Using the identity cos2x = 2cos^2x - 1, we can further simplify:
LHS = (1/2) * ((2cos^2a - 1) * cos^2b + (2cos^2b - 1) * sin^2a) - 1
= (1/2) * (2cos^2a * cos^2b - cos^2b + 2cos^2b * sin^2a - sin^2a) - 1
= cos^2a * cos^2b + cos^2b * sin^2a - 1/2 * cos^2b - 1/2 * sin^2a - 1
Now, we'll simplify the right-hand side (RHS) of the equation:
RHS = cos2acos2b
Using the identity cos2x = 2cos^2x - 1, we can rewrite the RHS:
RHS = (2cos^2a - 1) * (2cos^2b - 1)
= 4cos^2a * cos^2b - 2cos^2a - 2cos^2b + 1
Now that we have the LHS and RHS in simplified forms, let's compare them:
LHS = cos^2a * cos^2b + cos^2b * sin^2a - 1/2 * cos^2b - 1/2 * sin^2a - 1
RHS = 4cos^2a * cos^2b - 2cos^2a - 2cos^2b + 1
Simplifying further, we have:
LHS = cos^2a * cos^2b + cos^2b * sin^2a - 1/2 * cos^2b - 1/2 * sin^2a - 1
= cos^2b * (cos^2a + sin^2a) - 1/2 * (cos^2b + sin^2a) - 1
= cos^2b - 1/2 - 1
= cos^2b - 3/2
RHS = 4cos^2a * cos^2b - 2cos^2a - 2cos^2b + 1
= 4cos^2a * cos^2b - (2cos^2a + 2cos^2b - 1)
= cos^2a * cos^2b - 1/2
Comparing the LHS and RHS, we see that:
LHS = cos^2b - 3/2
RHS = cos^2a * cos^2b - 1/2
Since LHS and RHS are not equal, the given equation 2cos^2a*cos^2b + 2sin^2asin^2b - 1 = cos2acos2b is not true.