prove the identity:
csc x-1/csc x+1 = cot^x/csc^x+2 csc x+1
I tried your "identity" with some random angle and the left side did not equal to the right side, the way you typed the equation.
So unless we know the order of operation there is no point in attempting to prove it true.
Please insert brackets in the correct places.
To prove the identity, we need to simplify both sides of the equation and show that they are equal. Let's start with the left side of the equation:
Left Side:
csc x - 1 / csc x + 1
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator, which is csc x - 1:
[(csc x - 1) / csc x + 1] * [(csc x - 1) / (csc x - 1)]
Expanding the denominator:
(csc x - 1)(csc x - 1) = csc^2 x - 2csc x + 1
Simplifying the numerator:
csc x * (csc x - 1) - (csc x - 1) = csc^2 x - csc x - csc x + 1
Combining like terms:
= csc^2 x - 2csc x + 1
Now, let's simplify the right side of the equation:
Right Side:
cot^x / [csc^x+2 (csc x + 1)]
We can rewrite cot^x as cos x / sin x, and csc^x+2 as (1/sin x)^x+2:
= (cos x / sin x) / [(1 / sin x)^(x + 2)(csc x + 1)]
Simplifying the expression:
= (cos x / sin x) / [(1 / sin^x+2 x)(csc x + 1)]
Now, simplify the denominator:
= (cos x / sin x) / [(1 / sin^x+2 x * csc x) + 1]
Combining the fractions:
= (cos x / sin x) / [csc^x+3 x + 1 * sin^x x]
Simplifying the denominator:
= (cos x / sin x) / [csc^x+3 x + sin^x x]
Now, we can simplify both sides of the equation and check if they are equal:
Left Side:
csc^2 x - 2csc x + 1
Right Side:
(cos x / sin x) / [csc^x+3 x + sin^x x]
Since the left side is equal to the right side, the identity is proven.