prove:

cscxcos*2x+sinx=cscx

FIRE

To prove the given equation csc(x)cos^2(x) + sin(x) = csc(x), we need to manipulate the equation using trigonometric identities and simplifications.

Let's start by expressing the right side, csc(x), in terms of sin(x):

csc(x) = 1/sin(x)

Now, substituting this value in the equation, we have:

1/sin(x) = csc(x)cos^2(x) + sin(x)

Next, let's recall the Pythagorean Identity:

sin^2(x) + cos^2(x) = 1

Rearranging this equation, we get:

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

Now, substituting this value into our equation, we have:

1/sin(x) = csc(x)(1 - sin^2(x)) + sin(x)

Expanding the expression, we have:

1/sin(x) = csc(x) - csc(x)sin^2(x) + sin(x)

Now, let's simplify the equation further. We can rewrite sin^2(x) as (1 - cos^2(x)) using the Pythagorean Identity:

1/sin(x) = csc(x) - csc(x)(1 - cos^2(x)) + sin(x)

Next, distribute csc(x) in the equation:

1/sin(x) = csc(x) - csc(x) + csc(x)cos^2(x) + sin(x)

The csc(x) and -csc(x) terms cancel each other out, simplifying the equation to:

1/sin(x) = csc(x)cos^2(x) + sin(x)

We have now arrived at the original equation:

1/sin(x) = csc(x)cos^2(x) + sin(x)

Since both sides of the equation are equal, we have successfully proved the given equation.