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34 votes
Show that √(1-cos A/1+cos A) =cosec A - cot A​

User Kenneth Ito
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1 Answer

13 votes
13 votes

Hi there!


\sqrt{(1-cosA)/(1+cosA)} =

We can begin by multiplying by its conjugate:


\sqrt{(1-cosA)/(1+cosA)} * \sqrt{(1+cosA)/(1+cosA)} = \\\\\sqrt{((1-cosA)(1 + cosA))/((1+cosA)(1 + cosA))} =

Simplify using the identity:


1 - cos^2A = sin^2A


\sqrt{((1-cos^2A))/((1+cosA)^2)} =\\\\\sqrt{((sin^2A))/((1+cosA)^2)} =

Take the square root of the expression:


{(sinA)/(1+cosA) =

Multiply again by the conjugate to get a SINGLE term in the denominator:


{(sinA)/(1+cosA) * {(1-cosA)/(1-cosA) =\\

Simplify:


{(sinA(1-cosA))/(1-cos^2A) =

Use the above trig identity one more:


{(sinA(1-cosA))/(sin^2A) =

Cancel out sinA:


{((1-cosA))/(sinA) =

Split the fraction into two:


{(1)/(sinA) - (cosA)/(sinA) =

Recall:


1/sinA = cscA\\\\cosA/sinA = cotA

Simplify:


(1)/(sinA) + (cosA)/(sinA) = \boxed{cscA - cotA}

User Vasilis Vasilatos
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