Question

sintheta/1-costheta=cosectheta+cottheta​

Answer 1

Given :

$\\ \large \red \bigstar \rm \: \frac{sin \theta}{1 - cos \theta} = cosec \theta + cot \theta$

To Prove :

$\\ \large \boxed{ \blue \bigstar \sf{ \frac{sin \theta}{1 - cos \theta} = cosec \theta + cot \theta}}$

Solution:

LHS:

$\\ \bigstar \large \sf \: \frac{sin \theta}{1 - cos \theta}$

$\bf \: \bigstar \: multiplying \: numerator \: \: and \: \: denominator \: by \: \bf \\ \bf (1 + cos \theta)$

$\\ \implies \large \sf \: \frac{sin \theta}{1 - cos \theta} \times \frac{1 + cos \theta}{1 + cos \theta} \\ \\ \\ \implies \large\sf \: \frac{sin \theta(1 +cos \theta) }{(1 - cos \theta)(1 + cos \theta)} \\ \\ \\ \implies \large\sf \: \frac{sin \theta(1 + cos \theta)}{1(1 + cos \theta) -cos \theta(1 + cos \theta)} \\ \\ \\ \implies \sf \: \frac{sin \theta + sin \theta \times cos \theta}{1 + cos \theta - cos \theta - {cos}^{2} \theta} \\ \\ \\ \implies \large\sf \: \frac{sin \theta + sin \theta \times cos \theta}{1 + \cancel{ cos \theta - cos \theta} - {cos }^{2} \theta } \\ \\ \\ \implies \large\sf \: \frac{1 + cos \theta}{sin \theta} \times \frac{1}{sin \theta} + \frac{cos \theta}{sin \theta} \\ \\ \\ \implies \large\boxed{\sf{cosec \theta + cot \theta}}$

RHS

$\\ \implies \large \sf \: cosec \theta + cot \theta \:$

Hence we can say LHS=RHS

$\\ \large \boxed{ \sf{ \red \bigstar \: \frac{sin \theta}{1 - cos \theta} = cosec \theta + cot \theta}}$

Hence proved

Some related trigonometric values :

$\large\sf{ (1) sec \theta =\dfrac{1}{cos \theta}}$

$\large\sf{(2) tan \theta =\dfrac{sin \theta}{cos \theta}}$

$\large\sf{(3) cosec \theta =\dfrac{1}{sin \theta}}$

$\large\sf{(4) cot \theta =\dfrac{1}{tan \theta}}$

$\large\sf{(5) cot \theta=\dfrac{cos \theta}{sin \theta}}$