Question

Ctgt-sin2t=ctgt*cos2t

Answer 1

 

 

По тригониметрическому тождеству sin^2t + cos^2 t = 1

tgt*ctgt = 1

 

Answer 2

To solve the equation ctan(2t) = ccot(2t), where c is a constant, we can start by manipulating the trigonometric identities.

First, let's rewrite the equation using the definitions of tangent and cotangent in terms of sine and cosine:

c * (sin(2t) / cos(2t)) = c * (cos(2t) / sin(2t))

Next, let's simplify the equation by canceling out the common factor of c:

sin(2t) / cos(2t) = cos(2t) / sin(2t)

To further simplify this expression, we can cross-multiply:

sin^2(2t) = cos^2(2t)

Now, we can use the Pythagorean identity for trigonometric functions, which states that sin^2(x) + cos^2(x) = 1. Applying this identity to our equation, we have:

1 - cos^2(2t) = cos^2(2t)

Rearranging the terms, we get:

2 * cos^2(2t) = 1

Dividing both sides by 2, we have:

cos^2(2t) = 1/2

Taking the square root of both sides, we obtain:

cos(2t) = ±sqrt(1/2)

Since cosine is positive in the first and fourth quadrants, we can ignore the negative sign. Thus, we have:

cos(2t) = sqrt(1/2)

To find the values of t that satisfy this equation, we can take the inverse cosine (or arccosine) of both sides:

2t = arccos(sqrt(1/2))

Now, we solve for t:

t = (1/2) * arccos(sqrt(1/2))

Therefore, the solutions for the equation ctan(2t) = ccot(2t) are given by:

t = (1/2) * arccos(sqrt(1/2)) + n * π,

where n is an integer representing additional solutions.