What is the simplest equivalent trigonometric expression for csc^2 A - 1/cot A csc A?
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To find the simplest equivalent trigonometric expression for csc^2 A - 1/cot A csc A, we will use the trigonometric identities.
1. Start by writing the given expression: csc^2 A - 1/cot A csc A
2. Recall the trigonometric identities:
- csc^2 A = 1/sin^2 A
- cot A = 1/tan A = cos A/sin A
- csc A = 1/sin A
3. Substitute these identities into the expression:
csc^2 A - 1/(cos A/sin A) * (1/sin A)
4. Simplify the expression:
Using the reciprocal property of fractions, the expression becomes:
1/sin^2 A - sin A/(cos A * sin^2 A)
5. Combine the fractions by finding a common denominator:
The common denominator for the fractions is sin^2 A * cos A.
The expression becomes:
(1 - sin A * sin A)/(sin^2 A * cos A)
6. Simplify further:
Using the Pythagorean identity, sin^2 A + cos^2 A = 1, we replace (1 - sin A * sin A) with cos^2 A.
The expression becomes:
cos^2 A/(sin^2 A * cos A)
7. Cancel out the common factors:
The cos A in the numerator cancels with the cos A in the denominator.
The final simplest equivalent trigonometric expression is:
cos A/(sin^2 A) or sec A/(sin A)^3