Some inequality with complex numbers Is there a straightforward way to prove the following inequality:
$$|1 + k\big(\exp(it)-1\big)|\leq 1 $$
where $k\in(0,1)$ and $t \in \mathbb{R}$ (correction, see dxiv answer) with $|t| \leq 1$, other than writing the quantity into its real and imaginary parts and checking that they satisfy the required inequalities (which is long and seems inelegant) ?   

Look at it geometrically. Fixing $k$, the left side ranges over the circle with center $1-k$ and radius $k$. We see that this circle is internally tangent to the unit circle, so regardless of $k$ any point on the LHS must be contained in the unit circle. This means they are of magnitude $\leq 1$. 
