The greatest and least values of $$sin$$−$$1$$⁡$$x3+$$$$cos$$−$$1$$⁡$$x3$$ are

Question

The greatest and least values of $$sin$$−$$1$$⁡$$x3+$$$$cos$$−$$1$$⁡$$x3$$ are

Infinity Learn · Accepted Answer

We have,  $$sin$$−$$1$$⁡$$x3+$$$$cos$$−$$1$$⁡$$x3$$   $$=$$$$sin$$−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡$$x3$$−$$3sin$$−$$1$$⁡xcos−$$1$$⁡xsin−$$1$$⁡$$x+$$$$cos$$−$$1$$⁡x    $$=$$π$$38$$−$$3sin$$−$$1$$⁡xcos−$$1$$⁡x$$π$$$$2=$$π$$38$$−$$3$$$$π$$$$2$$$$π$$$$2$$−$$sin$$−$$1$$⁡xsin−$$1$$⁡x   $$=$$π$$38$$−$$3$$$$π$$$$24sin1$$⁡$$x+3$$$$π$$$$2sin$$−$$1$$⁡$$x2$$  $$=$$π$$38+3$$$$π$$$$2sin$$−$$1$$⁡$$x2$$−π$$2sin$$−$$1$$⁡$$x=$$π$$38+3$$$$π$$$$2sin$$−$$1$$⁡x−π$$42$$−$$3$$$$π$$$$332=$$π$$332+3$$$$π$$$$2sin$$−$$1$$⁡x−π$$42So$$, the least value of $$sin$$−$$1$$⁡$$x3+$$$$cos$$−$$1$$⁡$$x3$$ is π$$332$$      $$sin$$−$$1$$⁡x−π$$42$$≤$$3$$$$π$$$$42$$∴   Greatest value $$=$$π$$332+9$$$$π$$$$216$$×$$3$$$$π$$$$2=7$$$$π$$$$38$$

The greatest and least values of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ are

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The greatest and least values of sin−1⁡x3+cos−1⁡x3 are

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The greatest and least values of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ are