Find the Exact Value cos((7pi)/8)^2

Problem

cos2((7*π)/8)

Solution

1. Identify the power-reduction identity for cosine, which is cos2(θ)=(1+cos(2*θ))/2

2. Substitute the given angle θ=(7*π)/8 into the identity.

3. Simplify the argument of the cosine function by calculating 2⋅(7*π)/8=(7*π)/4

4. Evaluate the exact value of cos((7*π)/4) Since (7*π)/4 is in the fourth quadrant with a reference angle of π/4 the value is √(,2)/2

5. Substitute this value back into the expression to get (1+√(,2)/2)/2

6. Simplify the fraction by multiplying the numerator and the denominator by 2

Final Answer

cos2((7*π)/8)=(2+√(,2))/4

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