1 - tanx sinx = cos2x + sin²x
I think i finally created a proper iidentity.. I just need somebody to try and prove it!
Thanks
$\displaystyle LHS = 1 - (tanx \times sinx) $
$\displaystyle = 1 - \frac{sinx}{cosx} \times sinx $
$\displaystyle = 1-\frac{sin^2x}{cosx}$
$\displaystyle RHS = cos2x + sin^2x = cos^2x-sin^2x+sin^2x =cos^2x $
So the identity you have created cannot be equal.
however, if you change your LHS to :
$\displaystyle 1 - (tanx \times sinx \times cosx) $
$\displaystyle = 1 - (\frac{sinx}{cosx} \times sinx \times cosx) $
$\displaystyle = 1-(sinx \times sinx) = 1-(sin^2x) = cos^2x$
In this case, your RHS will be equal to LHS giving you a proper identity.