Question

if f(2tanx/1+tan^2x)=(1+cos2x)(sec^2x+2tanx)/2 then f(1)=​

Answer 1

SOLUTION

GIVEN

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

TO DETERMINE

The value of f(1)

EVALUATION

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

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = \frac{(2 {\cos}^{2} x)( { \sec}^{2}x + 2 \tan x )}{2} }$

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = {\cos}^{2} x( { \sec}^{2}x + 2 \tan x ) }$

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = {\cos}^{2} x.{ \sec}^{2}x + {\cos}^{2} x. 2 \tan x }$

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = {\cos}^{2} x. \frac{1}{ {\cos}^{2} x} +2 {\cos}^{2} x. \frac{ \sin x}{ \cos x} }$

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = 1 +2 {\cos}^{} x. \sin x}$

$\displaystyle \sf{ \implies \: f \bigg( \sin 2x \:\bigg) = 1 +\sin 2x }$

Putting y = sin 2x we get

$\displaystyle \sf{ \: f ( y \:) = 1 + y }$

Putting y = 1 we get

$\displaystyle \sf{ \implies \: f ( 1 \:) = 1 + 1 }$

$\displaystyle \sf{ \implies \: f ( 1 \:) = 2}$

FINAL ANSWER

$\displaystyle \sf{ \: f ( 1 \:) = 2}$

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$If 1/2 is a root of the quadratic equation x²-mx-5/4=0 , then the value of m is?

https://brainly.in/question/21062028

2.Find the roots of the quadratic equation 6x2 +5x + 1 =0 by method of completing the squares.

https://brainly.in/question/30446963

3. Write a quadratic equation whose root are -4 and -5

https://brainly.in/question/24154410