Establish the following identity.
1/1-cosx + 1/1+cosx = 2csc(squared)x
To establish the given identity, we need to manipulate the left-hand side of the equation and simplify it until it is equal to the right-hand side.
Let's start by finding a common denominator for the two fractions on the left-hand side.
The common denominator is (1 - cos x)(1 + cos x).
Now, let's rewrite the two fractions with the common denominator:
(1 + cos x) / [(1 - cos x)(1 + cos x)] + (1 - cos x) / [(1 - cos x)(1 + cos x)]
Next, let's combine the two fractions into a single fraction:
[(1 + cos x) + (1 - cos x)] / [(1 - cos x)(1 + cos x)]
Simplifying the numerator further:
[1 + cos x + 1 - cos x] / [(1 - cos x)(1 + cos x)]
The cos x and -cos x terms in the numerator cancel each other out:
[2] / [(1 - cos x)(1 + cos x)]
Now, let's simplify the denominator:
(1 - cos x)(1 + cos x) = 1 - cos^2 x
Using the trigonometric identity cos^2 x = 1 - sin^2 x, we can substitute the denominator:
(1 - sin^2 x)
Finally, let's simplify further:
[2] / [1 - sin^2 x]
Using the reciprocal identity for sine, which states that csc x = 1/sin x, we can rewrite the expression as:
2csc^2 x
Therefore, the left-hand side of the equation simplifies to 2csc^2 x, which is equal to the right-hand side.
Hence, the given identity is established.