problem.

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Joined: Aug 4, 2011

by hemangsarkar, Sep 10, 2012, 5:12 PM

Q) if $f\left(\frac{2\tan{x}}{1 + \tan^2x}\right) = \frac{(\cos2x + 1)(\sec^2x + 2\tan{x})}{2}$ for all real $x$ ,

find the value of $f(4)$

my solution :

we simplify the right hand side.

$\cos2x + 1 = 2\cos^2x$

$\sec^2{x} + 2\tanx = (\tan{x} + 1)^2$

so we have $f \left(\frac{2 \tan{x}}{1 + \tan^2x} \right) = (\cos^2x)(\tan{x} + 1)^2 = 1 + \sin2x$

$\frac{2 \tan{x}}{1 + \tan^2{x}} = \sin2x$

so, $f(m) = 1 + m$ for all $m$ .

hence $f(4) = 5$