Question

Sin^2π/13+cos^2π/13−(cos ^2π/12-sin ^2π/12)^2

Answer 1

Решение задания смотри на фотографии

Answer 2

Let's simplify the expression step by step:

1. sin^2(π/13) + cos^2(π/13): The trigonometric identity states that sin^2(x) + cos^2(x) = 1 for any angle x. So, in this case: sin^2(π/13) + cos^2(π/13) = 1.

2. cos^2(π/12) - sin^2(π/12): Again, using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the expression as follows: cos^2(π/12) - sin^2(π/12) = cos^2(π/12) - (1 - cos^2(π/12)) = 2cos^2(π/12) - 1.

3. Now we can substitute the result from step 1 and step 2 back into the original expression:

sin^2(π/13) + cos^2(π/13) - (cos^2(π/12) - sin^2(π/12))^2 = 1 - (2cos^2(π/12) - 1)^2 = 1 - (2cos^2(π/12))^2 + 2(2cos^2(π/12)) - 1 = 1 - 4cos^4(π/12) + 4cos^2(π/12) - 1.

4. Now, we can simplify further:

1 - 4cos^4(π/12) + 4cos^2(π/12) - 1 = 4cos^2(π/12) - 4cos^4(π/12).

At this point, we can't simplify any further without knowing the exact value of cos(π/12) and cos^4(π/12). If you have specific numerical values for these trigonometric functions, you can plug them in and calculate the result. Otherwise, this is the simplest form of the expression using trigonometric functions.