JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Inverse Trigonometric Functions

For

x \in (- 1, 1]

, the number of solutions of the equation

sin^{- 1} x = 2 tan^{- 1} x

is equal to ______.

Answer: 2

Solution

(- 1, 1]

both

sin^{- 1} x

and

2 tan^{- 1} x

pass through the origin and both reach

\frac{π}{2}

x = 1

, since

sin^{- 1} 1 = \frac{π}{2}

and

2 tan^{- 1} 1 = \frac{π}{2}

. Writing

g (x) = sin^{- 1} x - 2 tan^{- 1} x

, a small-

x

expansion gives

g (x) \approx - x + \frac{5}{6} x^{3}

, so

g

is negative on both sides of

0

and does not vanish for

0 < ∣ x ∣ < 1

. Thus the curves cross only at

x = 0

and at

x = 1

in the given interval, the point

x = - 1

being excluded. Therefore the number of solutions is

2

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