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

Answers (1)

D Deependra Verma

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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