Step 1: Check option (A): $\tan 45^{\circ}=1$ and $\cot 45^{\circ}=1$, so this is TRUE.
Step 2: Check option (B): $\sin 90^{\circ}=1$ and $\tan 45^{\circ}=1$, so this is TRUE.
Step 3: Check option (C): $\sin 30^{\circ}=\dfrac{1}{2}$ but $\cos 30^{\circ}=\dfrac{\sqrt{3}}{2}$.
$$\frac{1}{2}\neq\frac{\sqrt{3}}{2}$$
So this statement is FALSE.
Step 4: Check option (D): $\sin 45^{\circ}=\dfrac{1}{\sqrt{2}}=\cos 45^{\circ}$, so this is TRUE.
Correct option is (C).