Question

$Show space that space tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets equals straight pi over 4 plus 1 half cos to the power of negative 1 end exponent space straight x squared$ .

Solution

Put x² = cos θ

straight L. straight H. straight S. space equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets equals tan to the power of negative 1 end exponent space open square brackets fraction numerator square root of 1 plus cos space straight theta end root plus square root of 1 minus cos space straight theta end root over denominator square root of 1 plus cos space straight theta end root minus square root of 1 minus cos space straight theta end root end fraction close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent space open square brackets fraction numerator square root of 2 space cos squared begin display style straight theta over 2 end style end root plus square root of 2 space sin squared begin display style straight theta over 2 end style end root over denominator square root of 2 space cos squared begin display style straight theta over 2 end style end root minus square root of 2 space sin squared begin display style straight theta over 2 end style 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 cos begin display style straight theta over 2 end style plus square root of 2 sin begin display style straight theta over 2 end style over denominator square root of 2 cos begin display style straight theta over 2 end style minus square root of 2 sin begin display style straight theta over 2 end style end fraction close square brackets space space space space space space space space space space equals space tan to the power of negative 1 end exponent open square brackets fraction numerator cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style over denominator cos straight theta over 2 minus sin straight theta over 2 end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 plus tan begin display style straight theta over 2 end style over denominator 1 minus tan begin display style straight theta over 2 end style 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 plus straight theta over 2 close parentheses close square brackets space space space space space space space space space space equals space straight pi over 4 plus straight theta over 2 equals straight pi over 4 plus 1 half cos to the power of negative 1 end exponent straight x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space cos space straight theta equals straight x squared right square bracket space space space space space space space space space space equals straight R. straight H. straight S.

For Daily Free Study Material Join wiredfaculty

Inverse Trigonometric Functions

Some More Questions From Inverse Trigonometric Functions Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location