1) 8 cos x + 15 sin x = 17корень из 2/2 2) cos 5x*cos3x=1/2*cos 2x 3)sin 3x-sin17x=o

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1) To solve the equation 8 cos x + 15 sin x = 17√2/2, we can square both sides to eliminate the square root:

(8 cos x + 15 sin x)^2 = (17√2/2)^2
64 cos^2 x + 120 cos x sin x + 225 sin^2 x = 289/2

Using the trigonometric identity cos^2 x + sin^2 x = 1, we can rewrite the equation as:

64 - 64 sin^2 x + 120 cos x sin x + 225 sin^2 x = 289/2
64 - 64 sin^2 x + 120 cos x sin x + 225 sin^2 x = 144.5

Rearranging terms, we get:

-64 sin^2 x + 225 sin^2 x + 120 cos x sin x = 80.5

Combining like terms, we have:

161 sin^2 x + 120 cos x sin x = 80.5

This is a quadratic equation in terms of sin x. You can solve this equation by using the quadratic formula or by factoring.

2) To solve the equation cos(5x) cos(3x) = 1/2 cos(2x), you can use the angle addition formula for cosine:

cos(5x) cos(3x) = 1/2 [cos(5x+3x) + cos(5x-3x)]

Expanding the right side using the angle addition formulas:

cos(5x) cos(3x) = 1/2 [cos(8x) + cos(2x)]

Now, set the left side equal to the right side and solve for x.

3) To solve the equation sin(3x) - sin(17x) = 0, you can use the angle difference formula for sine:

sin(a) - sin(b) = 2 cos((a+b)/2) * sin((a-b)/2)

Using this formula, the equation becomes:

2 cos((3x+17x)/2) * sin((3x-17x)/2) = 0

Since sin(20x) = 0, the equation simplifies to:

cos(10x) = 0

Solving for x, you can find the values of x that satisfy the equation.