If sin y+cos y=x verify the identity

(cos y-sin y)^3=(2-x^2)^3/2

Question

If sin y+cos y=x verify the identity

(cos y-sin y)^3=(2-x^2)^3/2

anonymous · Answer

given sin y+cos y=x
square both sides and find the value of 2siny cos y

ybarrap · Answer

Start with $~\sin y+\cos y=x$ 

Square both sides: $(\sin y+\cos y)^2=\sin^2y+2\cos y\sin y+\cos^2=x^2\$

Next, given  $(\cos y-\sin y)^3=(2-x^2)^{3/2}\$, take cube roots on both sides: 
$\cos y-\sin y=(2-x^2)^{1/2}\$
Now square both sides:$(\cos y-\sin y)^{2}=2-x^2\$
Substitute $x^2$ derived above and simplify:

$\cos^2y-2\sin y\cos y+\cos^2y=2-\sin^2y-2\cos y\sin y-\cos^2$
$
1-2\sin y\cos y=2-2\cos y\sin y-1\
-2\sin y\cos y=-2\cos y\sin y
$

Which shows that $\sin y+\cos y=x$  is a solution to $(\cos y-\sin y)^3=(2-x^2)^{3/2}$ .