Maths-
General
Easy

Question

If c o s to the power of negative 1 end exponent invisible function application x minus c o s to the power of negative 1 end exponent invisible function application fraction numerator y over denominator 2 end fraction equals alpha comma then 4 x to the power of 2 end exponent minus 4 x y c o s invisible function application alpha plus y to the power of 2 end exponent text end textis equal to-

  1. 2 sin 2 alpha    
  2. 4    
  3. 4 s i n to the power of 2 end exponent invisible function application alpha    
  4. negative 4 s i n to the power of 2 end exponent invisible function application alpha    

hintHint:

cos to the power of negative 1 end exponent space x space minus space cos to the power of negative 1 end exponent space y space equals space cos to the power of negative 1 end exponent space left parenthesis space x y space plus space square root of 1 minus space x squared end root square root of 1 minus y squared end root right parenthesis space comma space comma space i f space x comma space y space greater than space 0 space a n d space x squared plus y squared less or equal than 1.

The correct answer is: 4 s i n to the power of 2 end exponent invisible function application alpha


    Given ; c o s to the power of negative 1 end exponent invisible function application x minus c o s to the power of negative 1 end exponent invisible function application fraction numerator y over denominator 2 end fraction equals alpha comma
    To Find : 4 x to the power of 2 end exponent minus 4 x y c o s invisible function application alpha plus y to the power of 2 end exponent text end text
    Detailed Solution
    We know that,
    cos to the power of negative 1 end exponent space x space minus space cos to the power of negative 1 end exponent space y space equals space cos to the power of negative 1 end exponent space left parenthesis space x y space plus space square root of 1 minus space x squared end root square root of 1 minus y squared end root right parenthesis space comma space comma space i f space x comma space y space greater than space 0 space a n d space x squared plus y squared less or equal than 1.
    Using above formula
    rightwards double arrow c o s to the power of negative 1 end exponent invisible function application x minus c o s to the power of negative 1 end exponent invisible function application y over 2 equals space cos to the power of negative 1 end exponent space left parenthesis space fraction numerator x y over denominator 2 end fraction space plus space square root of 1 minus space x squared end root square root of 1 minus y squared over 4 end root right parenthesis space equals space alpha

rightwards double arrow space fraction numerator x y over denominator 2 end fraction space plus space square root of 1 minus space x squared end root square root of 1 minus y squared over 4 end root space equals space c o s alpha

F u r t h e r space s i m p l i f y i n g

rightwards double arrow space space square root of 1 minus space x squared end root square root of 1 minus y squared over 4 space equals end root space space space c o s alpha space minus space fraction numerator x y over denominator 2 end fraction space

S i m p l i f y i n g space a n d space S q u a r i n g space o n space b o t h space s i d e s

rightwards double arrow space space left parenthesis 2 square root of 1 minus space x squared end root square root of 1 minus y squared over 4 space end root right parenthesis squared space equals space left parenthesis 2 c o s alpha space minus space x y right parenthesis squared

rightwards double arrow space space fraction numerator 4 left parenthesis 1 minus space x squared right parenthesis left parenthesis 4 minus space y squared right parenthesis over denominator 4 end fraction space equals space 4 c o s squared alpha space plus space x squared y squared space minus space 4 x y cos alpha

rightwards double arrow 4 space minus space y squared space minus 4 x squared space plus space x squared y to the power of 2 space end exponent plus space 4 x y cos alpha space minus space space x squared y squared space equals space 4 c o s squared alpha

rightwards double arrow space 4 space minus space 4 x squared space space minus space y squared space space plus space space 4 x y cos alpha space space equals space 4 c o s squared alpha

rightwards double arrow space 4 space minus space space 4 c o s squared alpha space equals space 4 x squared space plus space y squared space space space minus space space 4 x y cos alpha

space rightwards double arrow space 4 left parenthesis 1 space minus space space c o s squared alpha right parenthesis space equals space 4 x squared space plus space y squared space space space minus space space 4 x y cos alpha

rightwards double arrow space 4 sin squared alpha space space equals space 4 x squared space plus space y squared space space space minus space space 4 x y cos alpha

T h u s comma space space space 4 x squared space plus space y squared space space space minus space space 4 x y cos alpha space equals space 4 sin squared alpha

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