how to find the exact value of: cos(9*pi/12)*cos(5*pi/12)+sin(9*pi/12)*sin(5*pi/12)

9pi/12 should be converted into degrees. The value of pi is 180 so all you do is 180 times 9 divided by 12 and then take the cosine of that value.

cosx cosy + sinx siny = cos(x-y)

so, what we have here is

cos(9π/12 - 5π/12) = cos(4π/12) = cos π/3 = 1/2

To find the exact value of the expression cos(9*pi/12) * cos(5*pi/12) + sin(9*pi/12) * sin(5*pi/12), we can use the trigonometric identity known as the product-to-sum formula.

The product-to-sum formula states that

cos(A) * cos(B) + sin(A) * sin(B) = cos(A - B).

Therefore, to find the value of the given expression, we can find cos(9*pi/12 - 5*pi/12), which simplifies to cos(4*pi/12).

Now, we need to simplify the angle 4*pi/12. We can convert it to the smallest possible angle by dividing both the numerator and denominator by their greatest common divisor, which is 4.

4*pi/12 simplifies to pi/3.

So, the expression cos(9*pi/12) * cos(5*pi/12) + sin(9*pi/12) * sin(5*pi/12) is equivalent to cos(pi/3).

The exact value of cos(pi/3) is 1/2.

Therefore, the exact value of the given expression is 1/2.