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Verify the trigonometric identity: (cot A * cos A)/(cot A + cos A) = (cot A - cos A)/(cot A * cos A). Is this identity true?

1. True for all valid A
2. False
3. True only for A = 45 deg
4. Cannot be determined

Correct answer: True for all valid A

Solution

Cross-multiplying reduces the identity to cot² A * cos² A = cot² A - cos² A, which is true since cot² A - cos² A = cos² A(1/sin² A - 1) = cos² A * cos² A/sin² A = cot² A cos² A.

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