cotxcosx/csc^2x-1
Please help me verify
well, csc^2x-1 = cot^2x
cot*cos/cot^2
= cos/sin * cos * sin^2/cos^2
= sin
To verify the expression cot(x)cos(x) / (csc^2(x) - 1), we can simplify it step by step.
1. Let's start by rewriting cot(x) in terms of sin(x) and cos(x). The cotangent function is the reciprocal of the tangent function, so we have cot(x) = 1 / tan(x).
2. Next, we rewrite csc(x) as the reciprocal of sin(x). Thus, we have csc(x) = 1 / sin(x).
3. We substitute these expressions into the original expression:
cot(x)cos(x) / (csc^2(x) - 1)
= (1 / tan(x)) * cos(x) / ( (1 / sin(x))^2 - 1 )
4. Simplifying the denominator, we have:
= (1 / tan(x)) * cos(x) / (1/sin(x))^2 - 1
= (1 / tan(x)) * cos(x) / (1/sin(x)) * (1/sin(x)) - 1
= (1 / tan(x)) * cos(x) * sin^2(x) / (sin(x))^2 - 1
= cos(x) * sin^2(x) / (tan(x) * sin^2(x)) - 1
5. Now, we can simplify further by canceling out sin^2(x):
= cos(x) / tan(x) - 1
6. Using a trigonometric identity, we know that tan(x) = sin(x) / cos(x).
Therefore, cos(x) / tan(x) can be rewritten as cos(x) / (sin(x) / cos(x)).
Simplifying this further, we have cos^2(x) / sin(x).
7. Substituting this result back into the expression, we have:
= cos^2(x) / sin(x) - 1
So, the simplified form of the expression cot(x)cos(x) / (csc^2(x) - 1) is cos^2(x) / sin(x) - 1.