Cos (п/4-х) < корень из 2/2

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To solve this inequality, we first need to find the range of values for the given expression cos (π/4 - x).

Using the trigonometric identity cos (a - b) = cos a cos b + sin a sin b, we get:

cos (π/4 - x) = cos π/4 cos x + sin π/4 sin x

Since cos π/4 = sin π/4 = √2/2, we have:

cos (π/4 - x) = (√2/2)(√2/2) + (√2/2)(sin x)
cos (π/4 - x) = 2/4 + (√2/2)(sin x)
cos (π/4 - x) = 1/2 + (√2/2)(sin x)

Now, our inequality is:

1/2 + (√2/2)(sin x) < √2/2

Subtracting 1/2 from both sides, we get:

(√2/2)(sin x) < √2/2 - 1/2
(√2/2)(sin x) < 1/2

Dividing by √2/2, we get:

sin x < 1

This means that the inequality cos (π/4 - x) < √2/2 holds for all values of x where sin x < 1.

Therefore, the solution to the inequality cos (π/4 - x) < √2/2 is all real numbers x.