Question

Tan A-CotA÷Tan A + cot A​

Answer 1

`2\sin^2 A - 1`

Explanation:

`(\tan A - \cot A)/(\tan A + \cot A​) =`

`= ((\sin A)/(\cos A) - (\cos A)/(\sin A))/((\sin A)/(\cos A) + (\cos A)/(\sin A))`

`= (\sin A \cos A * ((\sin A)/(\cos A) - (\cos A)/(\sin A)))/(\sin A \cos A * ((\sin A)/(\cos A) + (\cos A)/(\sin A)))`

`= (\sin^2 A - \cos^2 A)/(\sin^2 A + \cos^2 A)`

`= (\sin^2 A - \cos^2 A)/(1)`

`= \sin^2 A - \cos^2 A`

Since

`\sin^2 A + \cos^2 A = 1`

we know

`\cos^2 A = 1 - \sin^2 A`

we now substitute to get

`= \sin^2 A - (1 - \sin^2 A)`

`= \sin^2 A - 1 + \sin^2 A)`

`= 2\sin^2 A - 1`