if cosa+sina=2cosathen prove cosa-sina=2sina

Question

if cosa+sina=√2cosa then prove cosa-sina=√sina

Answer 1

cos(a) + sin(a) = 2 cos(a)

 

Treating this as an equation, not an identity,

subtract cos(a) from both sides:

 

sin(a) = cos(a)

On Unit Circle, x = cos(a) and y = sin(a).

What points on Unit Circle have equal x and y coordinates?

The points that are also on y = x; which have angles of 45°+180°*n, n integer.

sin(a) = cos(a) = ±1/sqrt(2)

a = 45°+180° * n, n integer.

 

=====

 

cos(a) - sin(a) = 2 sin(a)

cos(a) = 3 sin(a) = ±3/sqrt(10)

a = arccos(3/sqrt(10)) + 180° * n, n integer.

 

=====

 

etc.

Answer 2

I'm confused, Rekha.  There is nothing true about the above.  For one thing

 

cos(a) + sin(a) ≠ (√2)cos(a)

 

Consider x = 0.

 

cos(0) = 1 and sin (0) = 0

 

so

 

1 + 0 ≠ (√2)

 

A true expression would be:

 

cos(a) + sin(a) = (√2)cos(a - (pi/4))

 

Similarly, if we consider the right-hand side of the expression, and evaluate at a = 0, then we get the ridiculous conclusion that

 

1 = 0

 

which it does not.

 

Please check again and make sure you copied the problem correctly.