IF CosX+SinX=?2.CosX P.T CosX-SinX=?2.SinX

Answer : Given : cosA+ sinA = (underroot2 )*cosA 

To prove : cosA - sinA = (underroot2 )*sinA

 

we have ,

=> (cosA+sinA)² + (cosA -sinA) = 2

=> ( (underroot 2 ) *(cosA) ) + (cosA-sinA)² = 2  { using given equation}

=> 2cos²A + (cosA-sinA)² = 2  

=> (cosA-sinA)² = 2 - 2cos²A

=> (cosA - sinA)² = 2 ( sin²A )    { using 1 - cos²A = sin²A }

 

taking the square root both the sides we get ,

 

=>  cosA- sinA = (underroot2 )*sinA

            Hence Proved