find the exact value of the following (show your work using exact values)
a) sin^2 45 + cos^2 45
b)2sin^2 45 + cos (2 * 45)
a) We know that sin^2(x) + cos^2(x) = 1 for any angle x. Therefore, sin^2(45) + cos^2(45) = 1.
b) Using the double angle identity for cosine, cos(2x) = cos^2(x) - sin^2(x). Plugging in x=45, we have cos(2 * 45) = cos^2(45) - sin^2(45).
We also know that sin^2(x) + cos^2(x) = 1. Rearranging this equation, we have sin^2(x) = 1 - cos^2(x). Plugging this into the previous equation, we have cos(2 * 45) = cos^2(45) - (1 - cos^2(45)).
Simplifying further, cos(2 * 45) = cos^2(45) - 1 + cos^2(45).
Since sin^2(45) + cos^2(45) = 1, we substitute sin^2(45) = 1 - cos^2(45) into the previous equation to get cos(2 * 45) = 2cos^2(45) - 1.
Now we can find the exact value of cos(2 * 45). Since cos^2(45) = (1/√2)^2 = 1/2, we substitute this value into the previous equation to find cos(2 * 45) = 2(1/2) - 1 = 1 - 1 = 0.