If sum of the elements in the range of $$f(x)=$$$$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$x+$$$$tan$$−$$1$$⁡$$x+$$$$tan$$−$$1$$⁡$$1x+$$$$sec$$−$$1$$⁡$$x+cosec$$−$$1$$⁡x is kπ, then $$k=$$

Question

If sum of the elements in the range of  $$f(x)=$$$$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$x+$$$$tan$$−$$1$$⁡$$x+$$$$tan$$−$$1$$⁡$$1x+$$$$sec$$−$$1$$⁡$$x+cosec$$−$$1$$⁡x is kπ, then $$k=$$

Infinity Learn · Accepted Answer

We know that (1) $$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$x=$$π$$2$$;∀x∈[−$$1$$,$$1$$] (2) $$tan$$−$$1$$⁡$$x+$$$$tan$$−$$1$$⁡$$1x=$$π$$2$$ if x≥$$0$$−π$$2$$ if x<$$0$$ (3) $$sec$$−$$1$$⁡(x)+cosec$$−$$1$$⁡$$(x)=$$π$$2$$;∀x∈$$($$−∞,−$$1$$]∪[$$1$$,∞$$)Only$$ $$-1$$ and $$1$$ satisfies the given $$equationf($$−$$1)=$$π$$2+$$−π$$2+$$π$$2=$$π$$2f(1)=$$π$$2+$$π$$2+$$π$$2=3$$$$π$$$$2$$∴ Range is π$$2$$,$$3$$$$π$$$$2$$ Sum of the elements in the range is $$2$$π⇒$$k=2Therefore$$, the correct answer is $$(4).

If sum of the elements in the range of $f (x) = \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x + \tan^{- 1} \frac{1}{x} + \sec^{- 1} x + {cosec}^{- 1} x is k π, then k =$

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If sum of the elements in the range of f(x)=sin−1⁡x+cos−1⁡x+tan−1⁡x+tan−1⁡1x+sec−1⁡x+cosec−1⁡x is kπ, then k=

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If sum of the elements in the range of $f (x) = \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x + \tan^{- 1} \frac{1}{x} + \sec^{- 1} x + {cosec}^{- 1} x is k π, then k =$