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Inverse Trigonometric Functions

Question
CBSEENMA12035707
Wired Faculty App

Prove that
tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus square root of 1 minus straight x end root end fraction close square brackets space equals space straight pi over 4 minus 1 half cos to the power of negative 1 end exponent straight x comma space space fraction numerator negative 1 over denominator square root of 2 end fraction less or equal than straight x less or equal than 1

Solution

We need to prove that
tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus square root of 1 minus straight x end root end fraction close square brackets equals space straight pi over 4 minus 1 half cos to the power of negative 1 end exponent straight X minus fraction numerator 1 over denominator square root of 2 end fraction less or equal than straight x less or equal than 1
Consider x = cos2t;
straight L. straight H. straight S. space equals space tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus cos 2 straight t end root minus square root of 1 minus cos 2 straight t end root over denominator square root of 1 plus cos 2 straight t end root plus square root of 1 minus cos 2 straight t end root end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 cost minus space square root of 2 sint over denominator square root of 2 cost plus square root of 2 sint end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus tant over denominator 1 plus tant end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator tan begin display style straight pi over 4 end style minus tant over denominator 1 plus tan begin display style straight pi over 4 end style cross times tant end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 minus straight t close parentheses close square brackets equals straight pi over 4 minus straight t equals straight pi over 4 minus 1 half cos to the power of negative 1 end exponent straight x equals straight R. straight H. straight S.

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