Answer:

here's the answer-

Step-by-step explanation:

Yes, that is correct!

To prove this identity, we can start with the equation:

cos(2x) = 2cos^2(x) - 1

Using this identity, we can expand the left-hand side of the equation:

cos^2 (pi/8) + cos^2 ((3pi)/8) + cos^2 ((5pi)/8) + cos^2 ((7pi)/8)

= 1/2 [2cos^2 (pi/8) + 2cos^2 ((3pi)/8) + 2cos^2 ((5pi)/8) + 2cos^2 ((7pi)/8)]

= 1/2 [cos(2(pi/8)) + cos(2(3pi/8)) + cos(2(5pi/8)) + cos(2(7pi/8)) + 4]

= 1/2 [cos(pi/4) + cos(3pi/4) + cos(5pi/4) + cos(7pi/4) + 4]

= 1/2 [(1/sqrt(2)) - (1/sqrt(2)) - (1/sqrt(2)) + (1/sqrt(2)) + 4]

= 1/2 [4]

= 2

Therefore, we have proven that:

cos^2 (pi/8) + cos^2 ((3pi)/8) + cos^2 ((5pi)/8) + cos^2 ((7pi)/8) = 2

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