Question

Prove that:
sin theta (1+tan theta )+cos theta (1+cot theta ) = sec theta + Cosec theta.​

Answer 1

$\huge \boxed{ \bf{Answer}}$


$\bf{To \: prove \: \div \sin \theta \: (1 + tan \theta) + cos \theta(1 + cosec \theta)}$ = sec theta + Cosec theta


$\boxed{ \bf{Proof}}$


L.H.S,

$= \sin \theta \: (1 + tan \theta) + cos \theta(1 + cosec \theta)$

$= sin \theta(1 + \frac{sin \theta}{cos \theta} ) + cos \theta \: (1 + \frac{cos \theta}{sin \theta} )$

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

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

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

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

$= sec \: \theta \: + cosec \: \theta$

$= R.H.S$


Hence proved.


$\boxed{ \bf{Used \: \: formulas}}$

$\bf{tan \: \theta \: = \frac{sin \theta}{cos \theta} }$

$\bf{cot \: \theta \: = \frac{cos \theta}{sin \theta} }$

$\bf{ {sin}^{2} \theta \: + {cos}^{2} \theta \: = 1}$

Answer 2

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ua answer is in attachment