Question

Cos^2(5p/8)-sin^2(3p/8)

Answer 1

Cos^2(5p/8)-sin^2(3p/8)=-√2/2

Answer 2

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Answer 3

We can use trigonometric identities to simplify this expression.

Recall the following identity:

cos(2θ) = cos²(θ) - sin²(θ)

Using this identity, we can rewrite the expression as:

cos²(5π/8) - sin²(3π/8) = cos(2(5π/8)) - cos(2(π/2 - 3π/8))

Now, recall the following identity:

cos(π - θ) = -cos(θ)

Using this identity, we can simplify the expression further:

cos²(5π/8) - sin²(3π/8) = cos(2(5π/8)) + cos(2(3π/8))

Now, we can use another trigonometric identity:

cos(θ + φ) = cos(θ)cos(φ) - sin(θ)sin(φ)

Using this identity, we can write:

cos(2(5π/8)) = cos(π/4 + 3π/8) = cos(π/4)cos(3π/8) - sin(π/4)sin(3π/8)

Similarly, we have:

cos(2(3π/8)) = cos(π/2 - π/8) = cos(π/2)cos(π/8) + sin(π/2)sin(π/8)

Now, we can substitute these values back into the expression:

cos²(5π/8) - sin²(3π/8) = [cos(π/4)cos(3π/8) - sin(π/4)sin(3π/8)] + [cos(π/2)cos(π/8) + sin(π/2)sin(π/8)]

Simplifying further using known values of sine and cosine:

cos²(5π/8) - sin²(3π/8) = [(√2/2)(√2cos(π/8)) - (√2/2)(√2sin(π/8))] + [0 + (1)(sin(π/8))]

Finally, we can simplify to get the answer:

cos²(5π/8) - sin²(3π/8) = cos(π/8) - sin(π/8) = (cos(π/4 - π/8)) = √2/2 ∙ √2/2 - √2/2 ∙ √2/2 = 0

Therefore, the value of cos²(5π/8) - sin²(3π/8) is equal to 0.