Find the exact value:
cos(pi/16)cos(3pi/16)-sin(pi/16)sin(3pi/16)
To find the exact value of cos(pi/16)cos(3pi/16) - sin(pi/16)sin(3pi/16), we can use the identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In this case, A = pi/16 and B = 3pi/16. So, let's rewrite the expression using this identity:
cos(pi/16)cos(3pi/16) - sin(pi/16)sin(3pi/16) = cos(pi/16 - 3pi/16)
Now, let's simplify the expression inside the cosine function:
pi/16 - 3pi/16 = -2pi/16 = -pi/8
So, the expression becomes:
cos(-pi/8)
Now, the cosine function is an even function, which means that cos(-x) = cos(x). Therefore, we can rewrite the expression as:
cos(pi/8)
Finally, the exact value of cos(pi/8) can be found by evaluating it using a calculator or by using the double-angle identity:
cos(pi/8) = sqrt((1 + cos(pi/4))/2)
Using the value of cos(pi/4) = sqrt(2)/2, we can substitute it into the formula:
cos(pi/8) = sqrt((1 + sqrt(2)/2)/2)
Simplifying further, we get:
cos(pi/8) = sqrt(2 + sqrt(2))/2
So, the exact value of the expression cos(pi/16)cos(3pi/16) - sin(pi/16)sin(3pi/16) is sqrt(2 + sqrt(2))/2.