Question

Prove that root 1-costheta/1+costheta = cosectheta-cottheta

Answer 1

Step-by-step explanation:

√1-cos theta/1+cos theta

=√(1-costheta)²/1-cos²theta

=√(1-costheta)²/sin²theta

=1-costheta/sintheta

=1/sintheta-costheta/sintheta

=cosec theta-cot theta

proved

Answer 2

$=\csc\theta-\cot\theta$ proved

Step-by-step explanation:

Consider the provided information.

$\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \csc\theta-\cot\theta$

Consider the Left hand side of the expression.

Rationalize the expression.

$=\sqrt{\frac{(1-\cos\theta)(1-\cos\theta)}{(1+\cos\theta)(1-\cos\theta)}}$

$=\sqrt{\frac{(1-\cos\theta)^2}{1-\cos^2\theta}}$

$=\sqrt{\frac{(1-\cos\theta)^2}{sin^2\theta}}$ (∴sin²θ=1-cos²θ)

$=\frac{1-\cos\theta}{sin\theta}$

$=\frac{1}{\sin\theta}-\frac{\cos\theta}{sin\theta}$

$=\csc\theta-\cot\theta$

Hence, proved

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