Question

Cos a / (cos a/2 + sin a/2) cпростити

Answer 1

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Answer 2

To simplify the expression cos(a) / (cos(a/2) + sin(a/2)), you can use the trigonometric identity:

cos(2θ) = 2 * cos^2(θ) - 1

Let's apply this identity to the denominator. Let θ = a/2:

cos(a/2) = 2 * cos^2(a/4) - 1

Now, we can rewrite the expression as:

cos(a) / (2 * cos^2(a/4) - 1 + sin(a/2))

Next, we'll use another trigonometric identity:

1 - cos(θ) = sin^2(θ/2)

Using this identity for θ = a/2:

1 - cos(a/2) = sin^2(a/4)

Now, the expression becomes:

cos(a) / (2 * cos^2(a/4) + sin^2(a/4))

We can further simplify the denominator using the Pythagorean trigonometric identity:

cos^2(θ) + sin^2(θ) = 1

Applying this identity to θ = a/4:

cos^2(a/4) + sin^2(a/4) = 1

So, the expression now simplifies to:

cos(a) / (2 * 1) = cos(a) / 2

The simplified expression is cos(a)/2.