Question

How do i prove......explain
sinA(1+tanA)+cosA(1+cotA)=secA+cscA

Answer 1

We have to prove that :

$\sin\alpha (1 + \tan\alpha ) + \cos \alpha (1 + \cot\alpha ) = \sec \alpha + \csc \alpha$

On taking LHS :

$\sin\alpha (1 + \frac{ \sin\alpha }{ \cos \alpha } ) + \cos \alpha (1 + \frac{ \cos\alpha }{ \sin\alpha } \\ \\ = > \frac{ \sin \alpha }{ \cos \alpha } ( \cos\alpha + \sin\alpha ) + \frac{ \cos \alpha }{ \sin \alpha } ( \sin \alpha + \cos\alpha ) \\ \\ = > ( \sin\alpha + \cos \alpha )( \frac{ \sin \alpha }{ \cos\alpha } + \frac{ \cos \alpha }{ \sin\alpha } ) \\ \\ = >( \sin\alpha + \cos\alpha ) ( \frac{ { \sin }^{2} \alpha + { \cos}^{2} \alpha }{ \sin \alpha \cos\alpha } ) \\ \\ = > ( \frac{ \sin\alpha + \cos\alpha }{ \sin \alpha \cos\alpha } ) \\ \\ As \: we \: know \: that: \: { \sin}^{2} \alpha + { \cos}^{2} \alpha = 1 \\ \\ = > \frac{1}{ \cos \alpha } + \frac{1}{ \sin\alpha } \\ \\ = > \sec\alpha + \csc\alpha = RHS \\ \\ HENCE \: PROVED$

Answer 2

LHS
sinA(1+tanA)+cosA(1+cotA)

sinA(1+(sinA/cosA)) + cosA(1+(cosA/sinA)) 
=(sinA/cosA)(cosA+sinA) + (cosA/sinA)(sinA+cosA) 
=(cosA+sinA)[(((SinA)^2+(cosA)^2)/sinA... 
=(cosA+sinA)(1/sinAcosA) 
=secA + cosecA 
=RHS