Question

(cos theta cosec theta - sin theta sec theta) / cos theta + sin theta = cosec theat - sec theat ​

Answer 1

$\;\;\underline{\textbf{\textsf{ Given:-}}}$

• $\sf{LHS = \dfrac{cos \theta \: cosec \theta - sin\theta \: sec\theta}{cos \theta + sin \theta} }$

• $\sf{RHS = cosec \theta - sec \theta }$

$\;\;\underline{\textbf{\textsf{ To Prove:-}}}$

• L.H.S = R.H.S

$\;\;\underline{\textbf{\textsf{ Proof :-}}}$

$\sf{LHS = \dfrac{cos \theta \: cosec \theta - sin\theta \: sec\theta}{cos \theta + sin \theta} }$

$\sf{ \longrightarrow \dfrac{cos \theta \dfrac{1}{sin \theta} - sin \theta \dfrac{1}{cos \theta} }{cos \theta + sin \theta} }$

$\sf{ \longrightarrow \dfrac{ \dfrac{cos \theta}{sin \theta} - \dfrac{sin \theta}{cos \theta} }{cos \theta + sin \theta} }$

$\sf{ \longrightarrow \dfrac{ \dfrac{ {cos}^{2} \theta - {sin}^{2} \theta }{sin \theta \: cos \theta} }{(cos \theta +sin \theta) } }$

$\sf{\longrightarrow \dfrac{ \dfrac{ \cancel{(cos \theta + sin\theta})(cos\theta - sin\theta}{sin\theta \: cos\theta} }{ \cancel{(cos \theta + sin\theta)} }}$

$\sf{ \longrightarrow \dfrac{cos \theta - sin \theta}{sin \theta \: cos \theta} }$

$\sf{ \longrightarrow \dfrac{1}{sin \theta} - \dfrac{1}{cos \theta} }$

$\sf{\longrightarrow cosec \theta - sec \theta }$

$\;\;\underline{\textbf{\textsf{ Hence -}}}$

LSH = RHS

( Proved)

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$\;\;\underline{\textbf{\textsf{Remember that :-}}}$

•$\sf{ cosec\theta = \dfrac{1}{sin\theta} }$

•$\sf{ sec\theta = \dfrac{1}{cos\theta} }$

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