Question

Sin(pi+a)*cos(pi-a)/ctg(3pi/2 -2a)

Answer 1

Решение
Sin(pi+a)*cos(pi-a)/ctg(3pi/2 -2a) = [- sina * (- cosa) ] / tg2a =
= [1/2 * (2sina * cosa)] / tg2a = (1/2 * sin2a) / tg2a = 
= { 1/2 * [ 2tga / (1 + tg²a)]} :  { 1/2 * [ 2tga / (1 - tg²a)] = (1 - tg²a) / (1 + tg²a) = = cos2a

Answer 2

To simplify the expression sin(π + a) * cos(π - a) / ctg(3π/2 - 2a), we can use trigonometric identities.

1. Recall that sin(π + a) = -sin(a) and cos(π - a) = -cos(a).
2. The cotangent (ctg) of an angle θ is equal to 1/tan(θ).

With these identities, we can rewrite the expression as follows:

sin(π + a) * cos(π - a) / ctg(3π/2 - 2a) = (-sin(a)) * (-cos(a)) / (1/tan(3π/2 - 2a))

Now, we can simplify further:

= (sin(a) * cos(a)) * tan(3π/2 - 2a)

Now, let's simplify tan(3π/2 - 2a):

tan(3π/2 - 2a) = tan(π/2 - 2a)

Since tan(π/2 - x) = 1/tan(x), we have:

tan(π/2 - 2a) = 1/tan(2a)

So, the simplified expression is:

(sin(a) * cos(a)) * (1/tan(2a))

You can leave it in this form, or if you'd like to simplify further, you can use the identity tan(2a) = 2tan(a) / (1 - tan^2(a)):

(sin(a) * cos(a)) * (1 / (2tan(a) / (1 - tan^2(a))))

Now, multiply by (1 - tan^2(a)):

(sin(a) * cos(a)) * ((1 - tan^2(a)) / (2tan(a)))

This is the simplified form of the expression sin(π + a) * cos(π - a) / ctg(3π/2 - 2a).