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

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