prove that: cos[tan^-1 {sin ( cot^-1 x)}] = {(1 + x^2)/(2 + x^2)}^1/2
Draw a right triangle with angle , "near side" x, and "opposite side" 1. That way "near side"/"opposite side" so = "opposite side"/"hypotenuse". The "opposite side" is 1 and, by the Pythagorean theorem, the "hypotenuse" is :
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Now, draw a right triangle with angle , "opposite side" 1 and "near side" so that and so .
is the "near side"/"hypotenuse" for this triangle. Since the "near side" is and the "opposite side" is 1, use the Pythagorean theorem again to find the hypotenuse of this triangle.
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