Find-value-s-of-x-if-sin-2cos-1-cot-2tan-1-x-0-

Question Number 24294 by ajfour last updated on 15/Nov/17

F i n d v a l u e (s) o f x i f

\sin [{2 \cos}^{- 1} {\cot ({2 \tan}^{- 1} x)}] = 0 .

Answered by mrW1 last updated on 15/Nov/17

\sin [{2 \cos}^{- 1} {\cot ({2 \tan}^{- 1} x)}] = 0

\Rightarrow {2 \cos}^{- 1} {\cot ({2 \tan}^{- 1} x)} = k π

\Rightarrow \cos^{- 1} {\cot ({2 \tan}^{- 1} x)} = \frac{k π}{2}, k \in [0, 2]

\Rightarrow \cot ({2 \tan}^{- 1} x) = \cos (\frac{k π}{2}), k \in [0, 2]

\Rightarrow \cot ({2 \tan}^{- 1} x) = \pm 1, 0

\Rightarrow {2 \tan}^{- 1} x = k π \pm \frac{π}{4}, k π + \frac{π}{2}

\Rightarrow \tan^{- 1} x = - \frac{3 π}{8}, - \frac{π}{4}, - \frac{π}{8}, \frac{π}{8}, \frac{π}{4}, \frac{3 π}{8}

\Rightarrow x = \pm \tan (\frac{π}{8}) = \pm (\sqrt{2} - 1)

\Rightarrow x = \pm \tan (\frac{π}{4}) = \pm 1

\Rightarrow x = \pm \tan (\frac{3 π}{8}) = \pm (\sqrt{2} + 1)

i . e . t h e r e a r e 6 v a l u e s f o r x .

Commented by ajfour last updated on 15/Nov/17

y e s s i r, t h a n k y o u .

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