Deducing $\sin x + \cos x \ge 1$ from $(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$ So in trig, say I have an acute angle $X$. And one can intuitively conclude that $\sin x + \cos x \ge 1$, but how does the fact that
$$(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$$
tell me that it is true that $\sin x + \cos x \ge 1$? I don't quite see the connection.
Thanks

If $x$ is acute $\sin x$ and $\cos x$ are both positive, and therefor
$$2\sin x\cos x\ge 0\\
1+2\sin x\cos x\ge 1\\
\sin^2x+\cos^2x+2\sin x\cos x\ge 1\\
\left(\sin x+\cos x\right)^2\ge 1$$
Where both $\sin x$ and $\cos x$ are positive, so is their addition:
$$\sin x+\cos x\ge 1$$
