Math, asked by aditya25112003, 1 year ago

sinxtan(x/2)+2cosx=2/1+tan^2(x/2)​

Answers

Answered by bansil003
8

\large{\underline{\boxed{\purple{ANSWER}}}}

TO PROVE:-

 \sin(x)  \tan( \frac{x}{2} )  + 2 \cos(x )  =   \frac{2}{1 +  \tan^{2} ( \frac{x}{2} ) }  \\

IDENTITIES TO BE USED:-

 \sin(2x)  =  \frac{2 \tan(x) }{1 +  \tan ^{2} (x) }

AND,

 \cos(2x)  =  \frac{1 -  \tan ^{2} (x) }{1 +  \tan ^{2} (x) }

\large{\underline{\boxed{\red{SOLUTION}}}}

ON L.H.S WE HAVE ,

 \sin( {x} )  \tan( \frac{x}{2} )  + 2 \cos(x)

So, using above identities , we get

  \tan( \frac{x}{2} ) ( \frac{2 \tan( \frac{x}{2} ) }{1 +  \tan ^{2} ( \frac{x}{2} ) } )   + 2( \frac{1 -  \tan ^{2} ( \frac{x}{2} ) }{1 +  \tan ^{2} ( \frac{x}{2} ) } ) \\  \\  \\  \\   =  >  \frac{2 \tan ^{2}  ( \frac{x}{2} ) - 2 \tan ^{2} ( \frac{x}{2} )   + 2} {1 +  \tan ^{2} ( \frac{x}{2} ) }  \\  \\  \\  \\  =  >  \frac{2}{1 +  \tan ^{2} ( \frac{x}{2} ) }

ALSO ON R.H.S , WE HAVE

 \frac{2}{1 +  \tan ^{2} ( \frac{x}{2} ) }  \\

HENCE ,

LHS = RHS

HENCE PROVED.

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