If cos\phi+sin\phi= \sqrt{2}cos\phi , then prove that cos?-sin?=?2sin?

Answers (1)

Solution:  We have , 

                               (cos phi +sin phi )= sqrt2cos phi                  squaring both side 

                                [(cos phi+sin phi)^2]= (sqrt2cos phi)^2

Rightarrow                            cos^2 phi+ sin^2 phi + 2cos phi. sin phi = 2 cos^2 phi

Rightarrow                            1 + 2 cos phi . sin phi = 2 cos^2 phi

Rightarrow                             cos phi . sin phi = cos^2 phi - frac12              .............................(1)

       From L.H.S ,        (cos phi - sin phi )^2= (cos phi + sin phi )^2 - 4 cos phi . sin phi                   

                                                                 = 2 cos^2phi - 4(cos^2 phi - frac12)                             put eq (1)

                                                                    = 2(1- cos^2 phi)=2sin^2 phi

Rightarrow                                          (cos phi - sin phi )^2 = 2 sin ^2 phi

                                                 (cos phi - sin phi )=sqrt2sinphi

Hence , it is proved .

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