Test Index

ICSE Class X Math 2014 Paper

© examsnet.com
Question : 51 of 52
Marks: +1, -0
Prove the identity:
(sin⁡heta+cos⁡heta)(anheta+cot⁡heta)(\sin heta + \cos heta)( an heta + \cot heta) =sec⁡heta+csc⁡heta= \sec heta + \csc heta
Solution:  
extL.H.S.=(sin⁡heta+cos⁡heta)(anheta+cot⁡heta)ext{L.H.S.} = (\sin heta + \cos heta)( an heta + \cot heta)
=(sin⁡θ+cos⁡θ)(sin⁡θcos⁡θ+cos⁡θsin⁡θ)= (\sin\theta + \cos\theta) \left( \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} \right)
=(sin⁡θ+cos⁡θ)(sin⁡2θ+cos⁡2θcos⁡θsin⁡θ)= (\sin \theta + \cos \theta) \left( \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta} \right)
=(sin⁡θ+cos⁡θ)×1cos⁡θsin⁡θ= (\sin \theta + \cos \theta) \times \frac{1}{\cos \theta \sin \theta}
=sin⁡θcos⁡θsin⁡θ+cos⁡θcos⁡θsin⁡θ= \frac{\sin\theta}{\cos\theta \sin\theta} + \frac{\cos\theta}{\cos\theta \sin\theta}
=1cos⁡θ+1sin⁡θ=sec⁡θ+csc⁡θ= \frac{1}{\cos \theta} + \frac{1}{\sin \theta} = \sec \theta + \csc \theta
=extR.H.S.= ext{R.H.S.}
© examsnet.com
Go to Question:
Prev Question
Next Question