Is it possible $\sin\alpha=\cos\alpha=1?$ Is it possible $\sin\alpha=\cos\alpha=1?$
We have defined trigonometry using a unit circle. The point $X(1;1)$ does not lie on the trig circle, right? Does this mean that $\alpha$ does not exist?
 A: $\sin ^2 x + \cos ^2 x = 1$ for all $x$, so if you had $\cos \alpha = \sin \alpha = 1$, then $\sin ^2 \alpha + \cos ^2 \alpha$ would be equal to $2$. This is absurd.
A: The answer you already gave is the clearest in my opinion. The trigonometric functions can be defined using the unit circle, so that $\sin(\theta)$ represents the $y$-coordinate when you have travelled $\theta$ degrees/radians anticlockwise around the unit circle, starting from the point $(1,0)$. Likewise, $\cos(\theta)$ represents the $x$-coordinate when you have travelled $\theta$ degrees/radians around.
Since the equation of a unit circle is $x^2+y^2=1$, and $(x,y)=(\cos(\theta),\sin(\theta))$, we obtain the identity
$$
\cos(\theta)^2+\sin(\theta)^2=1 \, ,
$$
which math's answer references. This is one way to answer your question. More directly, we know that the point $(1,1)$ does not lie on the unit circle, since the distance from the origin to that point is $\sqrt{2}$. Hence, it is never true that $\sin(\theta)=\cos(\theta)=1$.
