Question

If x = log (y +√y2 + 1 ) then y =

• A. tanh x
• B. coth x
• C. sinh x
• D. cosh x

Answer 1

The Correct Option is C

The correct option is: (C) sinh x.

y=sinh(x), let's start by solving the equation x=log(y+y2+1​) for y.

Given equation: x=log(y+y2+1​)

First, let's eliminate the logarithm by exponentiating both sides with base 10: 10x=y+y2+1​

Now, isolate the radical term on one side: 10x−y=y2+1​

Square both sides of the equation to eliminate the square root: (10x−y)2=y2+1

Expand the left-hand side: 102x−2xy+y2=y2+1

Now, simplify the equation: 102x−2xy=1

Isolate y on one side: 2xy=102x−1 y=2x102x−1​

Now, we'll express sinh(x) in terms of exponential functions: 2sinh(x)=2ex−e−x​

Comparing this expression with y=2x102x−1​, we notice a similarity if we substitute ex for 10x.

Therefore, we have: y=2xe2x−1​

Since y=2xe2x−1​ and sinh(x)=2ex−e−x​ are equivalent expressions, we can conclude that y=sinh(x).

Hence, the answer is justified: y=sinh(x).