Simplify csc(t)^2-sin(t)^2-cot(t)^2

Problem

csc^2(t)−sin^2(t)−cot^2(t)

Solution

1. Identify the Pythagorean identity relating the cosecant and cotangent functions.

csc^2(t)=1+cot^2(t)

2. Substitute the identity into the original expression to replace csc^2(t)

(1+cot^2(t))−sin^2(t)−cot^2(t)

3. Combine like terms by subtracting cot^2(t) from cot^2(t)

1−sin^2(t)

4. Apply the Pythagorean identity for sine and cosine, where 1−sin^2(t)=cos^2(t)

cos^2(t)

Final Answer

csc^2(t)−sin^2(t)−cot^2(t)=cos^2(t)

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