Question

(2cos^2(п/8)-1)/(1+8sin^2(п/8)cos^2(п/8))

Answer 1

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Answer 2

To simplify the expression (2cos^2(π/8) - 1)/(1 + 8sin^2(π/8)cos^2(π/8)), we can first replace π/8 with its numerical value and then use trigonometric identities to simplify it further.

Recall the following trigonometric identities:

1. cos(2θ) = 2cos^2(θ) - 1
2. sin^2(θ) + cos^2(θ) = 1

Now, let's proceed with the simplification:

1. Replace π/8 with its numerical value: π/8 = 3.14159/8 ≈ 0.3927.

2. Now, rewrite the expression using the trigonometric identities:

(2cos^2(π/8) - 1)/(1 + 8sin^2(π/8)cos^2(π/8))

= (cos(2 * (π/8)))/(1 + 8sin^2(π/8)cos^2(π/8))

= (cos(π/4))/(1 + 8sin^2(π/8)cos^2(π/8))

3. We can calculate cos(π/4):

cos(π/4) = 1/√2 ≈ 0.7071

4. Now, substitute this value back into the expression:

= 0.7071 / (1 + 8sin^2(π/8)cos^2(π/8))

5. Calculate sin(π/8):

sin(π/8) ≈ 0.3827

6. Substitute sin(π/8) into the expression:

= 0.7071 / (1 + 8(0.3827)^2 * cos^2(π/8))

7. Finally, we need to calculate cos(π/8):

cos(π/8) ≈ 0.9239

8. Substitute cos(π/8) into the expression:

= 0.7071 / (1 + 8(0.3827)^2 * 0.9239^2)

Now, calculate the value on the right-hand side to get the final result.