Cos A =3/4 find 16 cos^2 A/2 - 32 sin A/2 sin 5A/2?

Question

Answer

Given Information:

Calculation:

First, we need to find the value of sin A using the Pythagorean identity: sin^2 A + cos^2 A = 1
Given that cos A = 3/4, we have sin^2 A + (3/4)^2 = 1
sin^2 A + 9/16 = 1
sin^2 A = 1 - 9/16 = 7/16
sin A = sqrt(7)/4
Now, we can substitute the values of cos A and sin A into the given expression
16 cos^2(A/2) - 32 sin(A/2) sin(5A/2)
16(cos^2(A/2) - 2 sin(A/2) sin(5A/2))
Using the double angle formula for cosine and sine: cos(2x) = 2cos^2(x) - 1 and sin(2x) = 2sin(x)cos(x)
cos(A) = 2cos^2(A/2) - 1 and sin(5A) = 2sin(A/2)cos(A/2)
Substitute the values of cos A/2 and sin A/2 into the above formulas
cos(A) = 2(3/4)^2 - 1 = 3/2 - 1 = 1/2
sin(5A) = 2(sqrt(7)/4)(3/4) = 3sqrt(7)/8
Substitute these values back into the expression: 16(1/2) - 32(sqrt(7)/8)
8 - 4sqrt(7) = 8(1 - sqrt(7)/2)

Final Answer: