Kenneth S. answered • 04/21/16
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tan(cot-1 (-1/3) = tan (tan-1(-3) = - 3...because if cotangent is a fraction, then tangent is the reciprocal of that fraction.
Kenneth S.
Goodness, it does not appear that your teacher has prepared you properly for understanding inverse trigonometric functions.
The notation cot-1(x) means "what angle has its cotangent value equal to x"; this means that θ = cot-1x is equivalent to cot θ = x.
[This inverse function has another name: arccot.]
From Algebra II you should recall that if function f has an inverse f-1 then f(f-1(x)) = x.
Both the arctan and arccot functions have their ranges set as (-pi/2,pi/2); their arguments (domains) are any Real number.
Then we should know that tan θ = 1/x since cotangent and tangent functions are mutual reciprocals.
I said, in my answer: tan(cot-1 (-1/3) = tan (tan-1(-3) = - 3...because if cotangent is a fraction, then tangent is the reciprocal of that fraction.
What I did was to replace the cotangent inverse by tangent inverse of the reciprocal of -1/3. At that juncture, we have tan(tan-1(-3) and the tan of tan inverse causes cancellation of one another, leaving only the argument as the final answer, -3.
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04/21/16
Brian S.
04/21/16