tan((π/4)+(1/2) cos-1x)+ tan ((π/4)-(1/2) cos-1x),x ≠ 0 is equal to

Question

tan((π/4)+(1/2) cos-1x)+ tan ((π/4)-(1/2) cos-1x),x ≠ 0 is equal to (A) x (B) 2x (C) (2/x) (D) none of these

Tardigrade Admin · Accepted Answer

Let cos-1 x= θ. Then the given expression becomes ( 1+tan (θ/2) /1-tan (θ)2) +( 1- tan (θ/2)/1+tan (θ)2) = (2(1+tan 2 (θ/2))/1-tan2 (θ)2) = (2/ cos θ) = (2/x)

Q. $tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + t an (\frac{π}{4} - \frac{1}{2} cos^{- 1} x), x \neq = 0$ is equal to

$x$

$2 x$

$\frac{2}{x}$

none of these

Let $co s^{- 1} x = θ$ . Then the given expression becomes
$\frac{1 + t an \frac{θ}{2}}{1 - t an \frac{θ}{2}} + \frac{1 - t an \frac{θ}{2}}{1 + t an \frac{θ}{2}}$
$= \frac{2 ( 1 + t a n ^{2} \frac{θ}{2} )}{1 - t a n ^{2} \frac{θ}{2}}$
$= \frac{2}{cos θ}$
$= \frac{2}{x}$

Q. tan(4π​+21​cos−1x)+tan(4π​−21​cos−1x),x=0 is equal to

x

2x

x2​

none of these

Let cos−1x=θ. Then the given expression becomes 1−tan2θ​1+tan2θ​​+1+tan2θ​1−tan2θ​​ =1−tan22θ​2(1+tan22θ​)​ =cosθ2​ =x2​

Q. $tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + t an (\frac{π}{4} - \frac{1}{2} cos^{- 1} x), x \neq = 0$ is equal to

$x$

$2 x$

$\frac{2}{x}$

Let $co s^{- 1} x = θ$ . Then the given expression becomes
$\frac{1 + t an \frac{θ}{2}}{1 - t an \frac{θ}{2}} + \frac{1 - t an \frac{θ}{2}}{1 + t an \frac{θ}{2}}$
$= \frac{2 ( 1 + t a n ^{2} \frac{θ}{2} )}{1 - t a n ^{2} \frac{θ}{2}}$
$= \frac{2}{cos θ}$
$= \frac{2}{x}$