$$cot$$−$$1$$⁡$$($$$$cos$$⁡α$$)$$−$$tan$$−$$1$$⁡$$($$$$cos$$⁡α$$)=x$$, then $$sin$$ x $$=$$

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$$cot$$−$$1$$⁡$$($$$$cos$$⁡α$$)$$−$$tan$$−$$1$$⁡$$($$$$cos$$⁡α$$)=x$$⇒$$tan$$−$$1$$⁡$$1cos$$⁡α−$$tan$$−$$1$$⁡$$($$$$cos$$⁡α$$)=x$$⇒$$tan$$−$$1$$⁡$$1cos$$⁡α−$$cos$$⁡α$$1+1cos$$⁡α$$cos$$⁡α$$=x$$ ⇒$$tan$$−$$1$$⁡$$1$$−$$cos$$⁡α$$2cos$$⁡α$$=x$$⇒$$tan$$⁡$$x=1$$−$$cos$$⁡α$$2cos$$⁡α or $$cot$$⁡$$x=2cos$$⁡α$$1$$−$$cos$$⁡α or cosec⁡$$x=1+$$$$cos$$⁡α$$1$$−$$cos$$⁡α∴$$sin$$⁡$$x=1$$−$$cos$$⁡α$$1+$$$$cos$$⁡α$$=1$$−$$1$$−$$2sin2$$⁡α$$/21+2cos2$$⁡α$$/2$$−$$1$$ or $$sin$$⁡$$x=tan2$$⁡α$$/2$$

$\cot^{- 1} (\sqrt{\cos α}) - \tan^{- 1} (\sqrt{\cos α}) = x$ , then sin x =

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cot−1⁡(cos⁡α)−tan−1⁡(cos⁡α)=x, then sin x =

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$\cot^{- 1} (\sqrt{\cos α}) - \tan^{- 1} (\sqrt{\cos α}) = x$ , then sin x =