If y = tan-1 { (1 + x)1/2 – (1-x)1/2/ (1+ x)1/2 + (1- x)1/2

Prove that dy/dx = 1 / 2(1 – x2)1/2

Asked by haroonrashidgkp | 27th May, 2018, 12:57: PM

Expert Answer:

begin mathsize 16px style straight y equals 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
straight x space equals space cosθ space rightwards double arrow space straight theta equals cos to the power of negative 1 end exponent straight x
straight y equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus cosθ end root minus square root of 1 minus cosθ end root over denominator square root of 1 plus cosθ end root plus square root of 1 minus cosθ end root end fraction close square brackets
straight y 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 minus square root of 2 sin straight theta over 2 over denominator square root of 2 cos straight theta over 2 plus square root of 2 sin straight theta over 2 end fraction close square brackets
straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator cos straight theta over 2 minus sin straight theta over 2 over denominator cos straight theta over 2 plus sin straight theta over 2 end fraction close parentheses
straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus tan straight theta over 2 over denominator 1 plus tan straight theta over 2 end fraction close parentheses
straight y equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 minus straight theta over 2 close parentheses close square brackets
straight y equals straight pi over 4 minus straight theta over 2
straight y equals straight pi over 4 minus fraction numerator cos to the power of negative 1 end exponent straight x over denominator 2 end fraction
dy over dx equals 0 minus 1 half cross times fraction numerator negative 1 over denominator square root of 1 minus straight x squared end root end fraction
dy over dx equals fraction numerator 1 over denominator 2 square root of 1 minus straight x squared end root end fraction end style

Answered by Sneha shidid | 28th May, 2018, 10:56: AM