Question

simplify (cosec theta)(1+cos theta)(cosec theta -cot theta ) = 1

Answer 1

we know cosecA = 1/Sins
cot A= cosA/sinA
by this we solve it LHS

Answer 2

Answer:

$\to \sf{\red{cosec \theta (1+cos \theta)(cosec \theta - cot \theta)}}$

$\bullet \sf \ cosec \theta=\dfrac{1}{sin \theta} ; \ cot \theta = \dfrac{cos \theta}{sin \theta}$

$\big \therefore \sf \dfrac{1}{sin \theta} (1+cos \theta)(\dfrac{1}{sin \theta} - \dfrac{cos \theta}{sin \theta} )$

$\to \sf \dfrac{ (1+cos \theta)}{sin \theta} \left(\dfrac{1-cos \theta}{sin \theta} \right)$

$\to \sf \dfrac{ (1+cos \theta)(1-cos \theta)}{sin^2 \theta}$

$\to \sf \dfrac{(1-cos^2 \theta)}{sin^2 \theta}$

$\sf \bullet W.K.T: \ 1-cos^2 \theta = sin^2\theta$

$\to \sf \dfrac{(sin^2 \theta)}{sin^2 \theta}$

$\leadsto \sf{\red{1}}$

LHS=RHS.

HENCE PROVED.

Fundamental trigonometric identities:

$\boxed{\substack{\displaystyle \sf sin^2 \theta+cos^2 \theta = 1 \\\\ \displaystyle \sf 1+cot^2 \theta=cosec^2 \theta \\\\ \displaystyle \sf 1+tan^2 \theta=sec^2 \theta}}$