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  4. Number of values of x satisfying the equation cos -1(x2-. 5 x+6)=2 cot -1(1), is equal to

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Q. Number of values of $x$ satisfying the equation $\cos ^{-1}\left(x^{2}-\right.$ $5 x+6)=2 \cot ^{-1}(1)$, is equal to

Inverse Trigonometric Functions

Solution:

Given, $\cos ^{-1}\left(x^{2}-5 x+6\right)=2 \cot ^{-1} 1$
$\Rightarrow \cos ^{-1}\left(x^{2}-5 x+6\right)=2 \times \frac{\pi}{4}=\frac{\pi}{2} $
$\Rightarrow x^{2}-5 x+6=0$
$\Rightarrow(x-2)(x-3)=0$
$\therefore x=2,3$.
i.e. $2$ values