Question

1)10sin2x cos2x 2)tgx+tg4x/1-tgx*tg4x как спростить

Answer 1

$\mathtt{10sin2xcos2x=5*2sin2xcos2x=5cos4x}$

$\mathtt{\frac{tgx+tg4x}{1-tgxtg4x}=tg(x+4x)=tg5x}$

Answer 2

To simplify the expressions, let's work on each one separately:

1. Simplify 10sin(2x)cos(2x):

We can use the double-angle trigonometric identities to simplify this expression:

sin(2θ) = 2sin(θ)cos(θ) cos(2θ) = cos^2(θ) - sin^2(θ) = 1 - 2sin^2(θ) = 2cos^2(θ) - 1

So, for our expression 10sin(2x)cos(2x), let's use the identity sin(2θ) = 2sin(θ)cos(θ):

10sin(2x)cos(2x) = 10 * 2sin(x)cos(x) * (2cos^2(x) - 1)

Now, let's further simplify:

10 * 2sin(x)cos(x) * (2cos^2(x) - 1) = 20sin(x)cos(x) * (2cos^2(x) - 1)

Now, we can use the double-angle identity again, cos(2θ) = 2cos^2(θ) - 1:

20sin(x)cos(x) * (2cos^2(x) - 1) = 20sin(x)cos(x) * cos(2x)

Finally, using the identity sin(2θ) = 2sin(θ)cos(θ), we can write sin(x)cos(x) as 1/2sin(2x):

20sin(x)cos(x) * cos(2x) = 20 * 1/2sin(2x) * cos(2x)

Now, we have:

20 * 1/2sin(2x) * cos(2x) = 10sin(2x)cos(2x)

So, the simplified expression is 10sin(2x)cos(2x).

2. Simplify tg(x) + tg(4x) / (1 - tg(x) * tg(4x)):

Let's simplify the expression step by step:

First, we need to use the tangent sum identity, which states that:

tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) * tan(B))

In this case, A = x and B = 4x:

tg(x) + tg(4x) = tg(x + 4x) = tg(5x)

Now, the expression becomes:

tg(5x) / (1 - tg(x) * tg(4x))

Next, we need to use the tangent double-angle identity, which states that:

tan(2θ) = 2tan(θ) / (1 - tan^2(θ))

In this case, θ = 5x:

tg(5x) = tg(2 * 2.5x) = 2tg(2.5x) / (1 - tg^2(2.5x))

So, the final simplified expression is:

2tg(2.5x) / (1 - tg^2(2.5x))

This is the simplest form for the given expression.