Right triangle : Find $\angle{BPQ}$. $\triangle ABC, \angle{ABC} = 90^{\circ}$.
Let $P$ be the point on $BC$ such that $2 \cdot \angle{BAP} = \angle{CAP} = 14 ^{\circ}$ 
and $Q$ be the point on $AB$ such that $\angle{BCQ} = 23 ^{\circ}$. 
Find $\angle{BPQ}$.

My work :
$\angle{APB} = 83 ^{\circ}$
$\angle{BQC} = 67 ^{\circ}$
$\angle{ACQ} = 46 ^{\circ}$
Let $QC \cap AP= U$, $\angle{QUP} = 120 ^{\circ}$

Again, no need for trigonometric formulas and clumsy calculations. 

Let $D$ be the intersection point of $AP$ and $CQ$. By direct angle chasing, one can calculate that 
$$\angle \, ACD = \angle \, ACQ = 46^{\circ} = 2 \cdot 23^{\circ} = 2 \cdot \angle \,  BCQ = 2 \cdot \angle \, PCD$$
Moreover, observe that 
$$\angle \, CAD = \angle \, CAP = 14^{\circ} = 2 \cdot 7^{\circ} = 2 \cdot \angle \,  BAP = 2 \cdot \angle \, QAD$$
Let $I$ be the intersection point of the interior angle bisectors through vertices $A$ and $C$ of triangle $ACD$. Therefore, $DI$ is the interior angle bisector of angle $\angle \, ADC$. Again, some direct angle chasing leads to the conclusion that $\angle \, ADC = 120^{\circ}$ and $\angle \, QDA = \angle \, PDC = 60^{\circ}$. 
Now, observe that 
$$\angle \, QAD = 7^{\circ} = \frac{1}{2} \cdot 14^{\circ} = \frac{1}{2} \cdot \angle CAD = \angle \, IAD$$
$$\angle \, PCD = 23^{\circ} = \frac{1}{2} \cdot 46^{\circ} = \frac{1}{2} \cdot \angle \, ACD = \angle \, ICD$$
Furthermore, $$\angle \, IDA = \frac{1}{2} \cdot \angle \, ADC = \frac{1}{2} \cdot 120^{\circ} = 60^{\circ} = \angle  \, QDA$$
$$\angle \, IDC = \frac{1}{2} \cdot \angle \, ADC = \frac{1}{2} \cdot 120^{\circ} = 60^{\circ} = \angle  \, PDC$$ Consequently the triangles $ADQ$ and $ADI$ are congruent, as well as triangles $CDP$ and $CDI$ are also congruent. Therefore, $DQ = DI$ and $DP = DI$. Hence $DQ = DI = DP$ and so triangle $DPQ$ is isosceles with $\angle \, PDQ = 120^{\circ}$. Thus
$$\angle \, QPA = \angle \, QPD = 30^{\circ}$$ By angle chasing, $\angle \, BPA = 83^{\circ}$ and for that reason
$$\angle \, BPQ = \angle \, BPA - \angle \, QPA = 83^{\circ} - 30^{\circ} = 53^{\circ}$$  
