How do you do this?
1-sin^2x/csc^2 x-1
multipy numerator and denominator by sin^2
sin^2 (1-sin^2)/(1-sin^2)=sin^2 x
or, you could just convert it to
cos^2/cot^2 = cos^2 * tan^2
= cos^2*sin^2/cos^2 = sin^2
To simplify the expression (1 - sin^2x) / (csc^2x - 1), we can start by rewriting sin^2x and csc^2x using trigonometric identities.
Recall that the Pythagorean identity states that sin^2x + cos^2x = 1. Rearranging this equation, we can write sin^2x = 1 - cos^2x.
Now, csc x is the reciprocal of sin x, so we can rewrite it as 1/sin x. Squaring csc x, we get csc^2x = (1/sin x)^2 = 1/sin^2x.
Substituting the expressions for sin^2x and csc^2x into the original expression, we have:
(1 - (1 - cos^2x)) / (1/sin^2x - 1)
Next, simplify the numerator:
(1 - 1 + cos^2x) / (1/sin^2x - 1)
cos^2x simplifies to just cos^2x in the numerator:
cos^2x / (1/sin^2x - 1)
To simplify the denominator, we need to find a common denominator. The common denominator is sin^2x. Therefore, we have:
cos^2x / ((1 - sin^2x) / sin^2x)
Now, we can simplify further by multiplying the numerator by the reciprocal of the denominator:
cos^2x * (sin^2x / (1 - sin^2x))
Next, we can apply the identity sin^2x + cos^2x = 1:
(cos^2x * sin^2x) / (1 - sin^2x)
Finally, we have simplified the expression (1 - sin^2x) / (csc^2x - 1) to (cos^2x * sin^2x) / (1 - sin^2x).