Question #98d02

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Answer 1 · 2016-03-28T22:16:46

The given equality is not an identity.

Take $x = pi/4$ :

Left Hand Side $= sin^2(pi/4)+4sin(pi/4)+3/cos^2(pi/4)$

$= (sqrt(2)/2)^2+4(sqrt(2)/2)+3/(sqrt(2)/2)^2$

$=1/2 + 2sqrt(2) + 6$

$= 13/2+2sqrt(2)$

Right Hand Side $= sin(pi/4)+3/(1-sin(pi/4))$

$=sqrt(2)/2 + 3/(1-sqrt(2)/2)$

$=sqrt(2)/2 + (3(1+sqrt(2)/2))/(1/2)$

$=sqrt(2)/2 + (12+6sqrt(2))/2$

$=6+7/2sqrt(2)$

As the let hand side does not equal the right hand side at $x=pi/4$ , the given equality is not an identity.