Solve this
Question:

$\frac{\cos \theta \operatorname{cosec} \theta-\sin \theta \sec \theta}{\cos \theta+\sin \theta}=\operatorname{cosec} \theta-\sec \theta$

 

Solution:

$\mathrm{LHS}=\frac{\cos \theta \operatorname{cosec} \theta-\sin \theta \sec \theta}{\cos \theta+\sin \theta}$

$=\frac{\frac{\cos \theta}{\sin \theta}-\frac{\sin \theta}{\cos \theta}}{\cos \theta+\sin \theta}$

$=\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\cos \theta \sin \theta(\cos \theta+\sin \theta)}$

$=\frac{(\cos \theta+\sin \theta)(\cos \theta-\sin \theta)}{\cos \theta \sin \theta(\cos \theta+\sin \theta)}$

$=\frac{(\cos \theta-\sin \theta)}{\cos \theta \sin \theta}$

$=\frac{1}{\sin \theta}-\frac{1}{\cos \theta}$

$=\operatorname{cosec} \theta-\sec \theta$

$=$ RHS

Hence, LHS $=\mathrm{RHS}$

 

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