(3sin x + 4cos x)^2 + (3cos x - 4sin x)^2 =24

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вопрос

Let's expand the left side of the equation:

(3sin x + 4cos x)^2 = (3sin x)^2 + 2(3sin x)(4cos x) + (4cos x)^2 = 9sin^2 x + 24sin x cos x + 16cos^2 x,

(3cos x - 4sin x)^2 = (3cos x)^2 - 2(3cos x)(4sin x) + (4sin x)^2 = 9cos^2 x - 24sin x cos x + 16sin^2 x.

Adding the two expanded terms together, we get:

9sin^2 x + 24sin x cos x + 16cos^2 x + 9cos^2 x - 24sin x cos x + 16sin^2 x = 25(sin^2 x + cos^2 x) = 25.

Therefore, the given equation simplifies to:

24 = 24, which is true. So, the equation is satisfied for all values of x.