Question

1) arctg(x)+arcctg(3x)=π/2;2) arctg(x+3)+arcctg(4x)=π\2;​

Answer 1

Объяснение:

Второе по такой схеме,расписываешь арктангенс как пи на два минус арккотангенс

Answer 2

To solve the equations:

1. arctg(x) + arcctg(3x) = π/2

2. arctg(x+3) + arcctg(4x) = π/2

We can use trigonometric identities and properties of the arctan and arccot functions to simplify and solve these equations.

1. arctg(x) + arcctg(3x) = π/2

Let's start with the first equation:

arctg(x) + arcctg(3x) = π/2

We know that the sum of the arctan and arcctan of complementary angles is π/2. Therefore, we can rewrite the equation as:

arctg(x) + [π/2 - arctg(3x)] = π/2

Now, we can cancel out the π/2 on both sides:

arctg(x) - arctg(3x) = 0

Using the trigonometric identity for the difference of two arctan functions:

arctg(a) - arctg(b) = arctg((a - b) / (1 + ab))

In this case, a = x and b = 3x, so:

arctg(x - 3x) / (1 + x * 3x) = 0

Now, simplify:

arctg(-2x) / (1 + 3x^2) = 0

Now, we have arctan of something equal to 0, and the only way for that to happen is if the argument of arctan is 0:

-2x = 0

Solving for x:

x = 0

So, the solution to the first equation is x = 0.

Now, let's solve the second equation:

2. arctg(x+3) + arcctg(4x) = π/2

Using the same approach as before:

arctg(x+3) + [π/2 - arctg(4x)] = π/2

Cancel out the π/2 on both sides:

arctg(x+3) - arctg(4x) = 0

Now, use the identity for the difference of arctan functions:

arctg(a) - arctg(b) = arctg((a - b) / (1 + ab))

In this case, a = x+3 and b = 4x:

arctg((x + 3 - 4x) / (1 + (x+3) * 4x)) = 0

Simplify:

arctg(-3x - 3) / (1 + 4x(x+3)) = 0

Again, for arctan to equal 0, the argument must be 0:

-3x - 3 = 0

Solving for x:

-3x = 3

x = -1

So, the solution to the second equation is x = -1.

Therefore, the solutions to the given equations are:

1. x = 0

2. x = -1