Question

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

Answer 1

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}`