Question

prove Sin A + Cos A equal to root 2 cos 45 degree minus A​

Answer 1

Step-by-step explanation:

To prove ----->

SinA + CosA = √2 Cos ( 45° - A )

Proof -------> We know that ,

1) Sin45° = 1 / √2

2) Cos45° = 1 / √2

3) Cos ( A - B ) = CosA CosB - SinA SinB

Now returning to original problem,

LHS = SinA + CosA

Dividing and multiplying by √2 we get,

=> √ 2 ( 1/√2 SinA + 1 / √2 CosA )

Applying Sin45° = Cos45° = 1/√2 , we get,

=> √2 ( Sin45° SinA + Cos45° CosA )

=> √2 ( CosA Cos45° + SinA Sin45° )

Applying above identity , we get ,

=> √2 Cos ( 45° - A ) = RHS

Additional information----->

1) Cos ( A + B ) = CosA CosB - SinA SinB

2) Sin ( A + B ) = SinA CosB + CosA SinB

3) Sin ( A - B ) = SinA CosB - CosA SinB

4) tan( A + B ) = ( tanA + tanB ) / ( 1 - tanA tanB )

5) tan( A - B ) = ( tanA - tanB ) / ( 1 + tanA tanB )