If tan = ( 1 + 2 - x ) - 1 and tan = ( 1 + 2 x + 1 ) - 1 , then equals

Question

If $\tan α = {(1 + 2^{- x})}^{- 1}$ and $\tan β = {(1 + 2^{x + 1})}^{- 1},$ then $α + β$ equals

Smart Achievers · Accepted Answer

(2)

We know,

$\tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β}$

$∵$ $\tan α = \frac{1}{1 + 2^{- x}}$

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

$∴$ $\tan$ $(α + β)$ $= \frac{\frac{1}{1 + \frac{1}{2^{x}}} + \frac{1}{1 + 2^{x + 1}}}{1 - \frac{1}{1 + \frac{1}{2^{x}}} \cdot \frac{1}{1 + 2^{x + 1}}}$

$\Rightarrow \tan (α + β) = \frac{2^{x} + 2 \cdot 2^{2 x} + 2^{x} + 1}{1 + 2^{x} + 2 \cdot 2^{x} + 2 \cdot 2^{2 x} - 2^{x}}$

$\Rightarrow \tan (α + β) = 1$

$\Rightarrow$ $α + β = \frac{π}{4}$

Solution

Q.
If $\tan α = {(1 + 2^{- x})}^{- 1}$ and $\tan β = {(1 + 2^{x + 1})}^{- 1},$ then $α + β$ equals

1 $\frac{π}{6}$

2 $\frac{π}{4}$

3 $\frac{π}{3}$

4 $\frac{π}{2}$

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Solution

Q. If tanα=(1+2-x)-1 and tanβ=(1+2x+1)-1, then α+β equals

1 π6

2 π4

3 π3

4 π2