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fraction numerator d over denominator d x end fraction open parentheses Tan to the power of negative 1 end exponent invisible function application square root of fraction numerator 1 minus c o s invisible function application x over denominator 1 plus c o s invisible function application x end fraction end root close parentheses equals

  1. 1    
  2. fraction numerator 1 over denominator 2 end fraction    
  3. fraction numerator 1 over denominator 1 plus x to the power of 2 end exponent end fraction    
  4. fraction numerator negative 1 over denominator 1 plus x to the power of 2 end exponent end fraction    

hintHint:

We are given a function. We have to find it's derivative. Before finding the derivative, we will simplify the given function.

The correct answer is: fraction numerator 1 over denominator 2 end fraction


    The given function is fraction numerator d over denominator d x end fraction open parentheses tan to the power of negative 1 end exponent square root of fraction numerator 1 space minus space cos x over denominator 1 space plus space cos x end fraction end root close parentheses
    We will first simplify the given function.
    Let us denote the given function by y.
    W e space w i l l space ta k e space t h e space c o m p o n e n t space o f space t h e space f u n c t i o n space a n d space s i m p l i f y space i t
fraction numerator 1 space minus space cos x over denominator 1 space plus space cos x end fraction space equals fraction numerator 2 sin squared begin display style x over 2 end style over denominator 2 cos squared begin display style x over 2 end style end fraction space space space space space space space space... space open curly brackets table attributes columnalign left end attributes row cell cos 2 theta space equals space 1 space minus space 2 sin squared theta end cell row cell cos 2 theta space equals space 2 cos squared theta space minus space 1 end cell end table close
space space space space space space space space space space space space space space space space space space space equals space tan squared x over 2
    We will substitute this value in the above function.
    y space equals fraction numerator d over denominator d x end fraction tan to the power of negative 1 end exponent square root of tan squared x over 2 end root space
space space space equals fraction numerator d over denominator d x end fraction tan to the power of negative 1 end exponent left parenthesis tan x over 2 right parenthesis
space space space space equals fraction numerator d over denominator d x end fraction x over 2
space space space equals 1 half
S o comma space t h e space a n s w e r space o f space t h e space s o l u t i o n space i s space 1 half
space space space

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