How do you verify the identity #sec^2(x/2)= (2secx + 2)/(secx + 2 + cosx)#?

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Answer 1 · 2015-07-22T12:48:16

Required to prove : #sec^2(x/2)= (2secx + 2)/(secx + 2 + cosx)#

#"Right Hand Side"= (2secx + 2)/(secx + 2 + cosx)#

Remember that #secx=1/cosx#

#=> (2*1/cosx + 2)/(1/cosx + 2 + cosx)#

Now, multiply top and bottom by #cosx#

#=>(cosx xx(2*1/cosx + 2))/(cosx xx(1/cosx + 2 + cosx))#

#=>(2+2cosx)/(1+2cosx+cos^2x)#

Factorize the bottom,

#=>(2(1+cosx))/(1+cosx)^2#

#=>2/(1+cosx)#

Recall the identity : #cos2x=2cos^2x-1#
#=>1+cos2x=2cos^2x#

Similarly : #1+cosx=2cos^2(x/2)#

#=>"Right Hand Side"=2/(2cos^2(x/2))=1/cos^2(x/2)=color(blue)(sec^2(x/2)) = "Left Hand Side"#

As Required