Question

( cosec theta - sin theta )(sec theta - cos theta )( tan theta + cot theta ) =1

Answer 1

HOPE THIS HELPS YOU ☺️

Answer 2

Answer with step-by-step explanation:

To prove: $(cosec \theta - sin\theta)(sec\theta -cos\theta)(tan\theta+ cot\theta)=1$

L.H.S

$(cosec \theta - sin\theta)(sec\theta -cos\theta)(tan\theta+ cot\theta)\\\\\Rightarrow (\dfrac{1}{sin\theta}-sin\theta )(\dfrac{1}{cos\theta} -cos\theta)(tan\theta +\dfrac{1}{tan\theta})\\\\\Rightarrow (\dfrac{1-sin^2\theta}{sin\theta} )(\dfrac{1-cos^2\theta}{cos\theta})(\dfrac{1+tan^2\theta}{tan\theta} )$

Now as we know

$1- sin^2\theta= cos^2\theta , \\1-cos^2\theta=sin^2\theta ,\\1+tan^2\theta = sec^2\theta$

Therefore

$\Rightarrow (\dfrac{cos^2\theta}{sin\theta} )(\dfrac{sin^2\theta}{cos\theta} )(\dfrac{sec^2\theta}{tan\theta} )\\\\\Rightarrow (cos\theta )(sin\theta)(\dfrac{\dfrac{1}{cos^2\theta} }{\dfrac{sin\theta}{cos\theta} } )\\\\\\\Rightarrow (cos\theta )(sin\theta) \dfrac{1}{cos\theta sin\theta} =1$

= R.H.S.

Hence proved