The number of solution(s) of the equationsin(-1)2x - cos(-1)x + tan(-1)2x = pi/2is
(1) Zero (2) One(3) Two (4) Infinitely many

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Solution

$- 1 \leq 2 x \leq 1 \cap x \in [- 1, 1] \cap 2 x \in R$
$\frac{- 1}{2} \leq x \leq \frac{1}{2} \cap - 1 \leq x \leq 1 \cap x \in R$
$\Rightarrow x \in [\frac{- 1}{2}, \frac{1}{2}]$
$- c o s^{- 1} x - t a n^{- 1} 2 x = \frac{π}{2} - s i n^{- 1} 2 x = c o s^{- 1} 2 x$
$- t a n^{- 1} 2 x = c o s^{- 1} 2 x + c o s^{- 1} x$
Converting $t a n^{- 1} t o c o s^{- 1}$ ,

$\Rightarrow - c o s^{- 1} \frac{\sqrt{4 x^{2} + 1}}{1} = c o s^{- 1} (2 x^{2} - \sqrt{1 - 4 x^{2}} \sqrt{1 - x^{2}})$
LHS is always -ve, and RHS is always +ve
$∴ c o s^{- 1} x \in [0, π]$
So, number of solution is zero.

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