Question

prove that : cosA/1+sinA+ 1-sinA/cosA =2 cosec A
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Answer 1

Let us start with the LHS  

LHS = 1 + cosA/sinA  +  sinA / 1+ cosA

here , LCM = sinA(1+ cosA)

=  \frac{(1+cosA) ^{2}  + (sinA) ^{2} }{sinA(1 + cosA)}  

=  \frac{ 1 + 2cosA + co s^{2}A + sin ^{2}A  }{sinA(1 + cosA)}  

We know that ,

sin²A + cos²A = 1  

Hence,

=  \frac{1 + 1 + 2 cosA}{sinA( 1 + cosA)}  

\frac{2 + 2 cosA}{sinA(1+cosA)}  

=  \frac{2 ( 1 + cosA)}{sinA(1+ cosA)}  

1 + cosA gets cancelled , ( in numerator and denominator).

= 2 / sinA

= 2* 1/sinA

We know that , 1/sinA  = cosec A

Hence,

= 2*1/sinA = 2* cosecA = 2cosecA = RHS

PROVED.