The expression \frac{tanA}{1 - cotA} + \frac{cotA}{1

Engineering

Mathematics

Definition of Trigonometric Ratio

Question

The expression $\frac{tanA}{1 - cotA} + \frac{cotA}{1 - tanA}$ can be written as :

sin A cos A + 1

sec A + cosec A

tan A + cot A

sec A cosec A + 1

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$\frac{tanA}{1 - cotA} + \frac{cotA}{1 - tanA} = \frac{tanA}{1 - \frac{1}{tanA}} + \frac{cotA}{1 - tanA} = \frac{\tan^{2} A}{tanA - 1} + \frac{cotA}{1 - tanA}$

$= \frac{\tan^{2} A - cotA}{tanA - 1} = \frac{\tan^{3} A - 1}{tanA (tanA - 1)} = \frac{(tanA - 1) (\tan^{2} A + tanA + 1)}{tanA (tanA - 1)}$

$= \frac{sinA}{cosA} + \frac{cosA}{sinA} + 1 = \frac{\sin^{2} A + \cos^{2} A}{cosAsinA} + 1 = secAcosec A + 1$