Question

if f ( 2 tan x /1+ tan square x ) = (1+ cos 2x ) ( sec square x +2 tan x ) / 2 then f ( 1/3) =​

Answer 1

The answer to this problem is 4/3

Now, assuming that what you meant to ask was

$f( \frac{ \tan(x) }{1 + { \tan(x) }^{2} } ) = \frac{ (1 + \cos(2x) )( { \sec(x) }^{2} + 2 \tan(x) )}{2}$

I am assuming that this is what you wanted to ask.

Using rules of trigonometry we arrive to this final equation.

$f(2 \sin(x) \cos(x)) = 1 + 2 \sin(x) \cos(x)$

(I am leaving the trigonometric manipulation up to you.)

Putting

$2 \sin(x ) \cos(x) = \sin(2x) = \frac{1}{3}$

$f( \frac{1}{3} ) = 1 + \frac{1}{3} = \frac{4}{3}$

I hope tis helps.

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