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.