Question

Sin^2(15Pi/4-2x)-cos^2(17Pi/4-2x)= упростить. с решением

Answer 1

To simplify the expression sin^2(15Pi/4-2x) - cos^2(17Pi/4-2x) and find its solution, we can use trigonometric identities and properties. Let's break it down step by step:

Step 1: Simplify the angles inside the trigonometric functions. - 15Pi/4 - 2x can be simplified to 7Pi/4 - 2x. - 17Pi/4 - 2x can be simplified to 9Pi/4 - 2x.

Step 2: Apply the trigonometric identity sin^2θ - cos^2θ = -cos(2θ). - sin^2(7Pi/4 - 2x) - cos^2(9Pi/4 - 2x) = -cos(2(7Pi/4 - 2x)) - cos(2(9Pi/4 - 2x)).

Step 3: Simplify the angles inside the cosine function. - 2(7Pi/4 - 2x) can be simplified to 14Pi/4 - 4x, which is equivalent to 7Pi/2 - 4x. - 2(9Pi/4 - 2x) can be simplified to 18Pi/4 - 4x, which is equivalent to 9Pi/2 - 4x.

Step 4: Apply the trigonometric identity cos(2θ) = cos^2θ - sin^2θ. - -cos(2(7Pi/4 - 2x)) - cos(2(9Pi/4 - 2x)) = -(cos^2(7Pi/2 - 4x) - sin^2(7Pi/2 - 4x)) - (cos^2(9Pi/2 - 4x) - sin^2(9Pi/2 - 4x)).

Step 5: Simplify the expression further. - -(cos^2(7Pi/2 - 4x) - sin^2(7Pi/2 - 4x)) - (cos^2(9Pi/2 - 4x) - sin^2(9Pi/2 - 4x)) = -cos^2(7Pi/2 - 4x) + sin^2(7Pi/2 - 4x) - cos^2(9Pi/2 - 4x) + sin^2(9Pi/2 - 4x).

Step 6: Apply the trigonometric identity sin^2θ + cos^2θ = 1. - -cos^2(7Pi/2 - 4x) + sin^2(7Pi/2 - 4x) - cos^2(9Pi/2 - 4x) + sin^2(9Pi/2 - 4x) = -(1 - cos^2(7Pi/2 - 4x)) + (1 - cos^2(9Pi/2 - 4x)).

Step 7: Simplify the expression further. - -(1 - cos^2(7Pi/2 - 4x)) + (1 - cos^2(9Pi/2 - 4x)) = -1 + cos^2(7Pi/2 - 4x) + 1 - cos^2(9Pi/2 - 4x).

Step 8: Combine like terms. - -1 + cos^2(7Pi/2 - 4x) + 1 - cos^2(9Pi/2 - 4x) = cos^2(7Pi/2 - 4x) - cos^2(9Pi/2 - 4x).

Therefore, the simplified expression is cos^2(7Pi/2 - 4x) - cos^2(9Pi/2 - 4x).

Unfortunately, without specific values for x, we cannot find a numerical solution for this expression. However, we have simplified it as much as possible.