Verify the identity.
csc t − sin t/csc t= cos2 t
You are missing a parenthesis !!!
I assume that you mean:
(csc t - sin t)/csc t
(1/sin t - sin t) /(1/sin t)
(1 - sin^2 t) / 1
cos^2 t
To verify the given identity:
csc(t) - sin(t) / csc(t) = cos^2(t)
We can start by manipulating the left side of the equation:
1. Rewrite csc(t) as 1/sin(t):
(1/sin(t)) - sin(t) / (1/sin(t))
2. Combine the fractions:
(1 - sin^2(t)) / sin(t)
Now, we know that sin^2(t) + cos^2(t) = 1 (from the Pythagorean Identity). Rearranging this equation gives us sin^2(t) = 1 - cos^2(t).
3. Substitute sin^2(t) with 1 - cos^2(t):
(1 - cos^2(t)) / sin(t)
4. Rewrite 1 as sin^2(t) + cos^2(t):
(sin^2(t) + cos^2(t) - cos^2(t)) / sin(t)
5. Simplify:
sin^2(t) / sin(t)
Finally, we can simplify further by canceling out common factors:
sin(t)
Therefore, the left side of the equation simplifies to sin(t), which is equal to the right side (cos^2(t)). Hence, the identity is verified.