Answer:
Use Cosine Sum identity
Step-by-step explanation:
It's a simple demonstration if you know what trigonometric identity you should use. In this case use cosine sum identity which states:
\(cos(a+b)=cos(a)*cos(b)-sin(a)*sin(b)\)
Also keep in mind this property of fractions:
\(\frac{a \pm b}{c} =\frac{a}{c} \pm \frac{b}{c} \hspace{3} c
eq0\)
Using the previous information:
\(\frac{cos(x+y)}{cos(x)*sin(y)}=\frac{cos(x)*cos(y)-sin(x)*sin(y)}{cos(x)*sin(y)}=\frac{cos(x)*cos(y)}{cos(x)*sin(y)}-\frac{sin(x)sin(y)}{cos(x)*sin(y)}\\ \\=\frac{cos(y)}{sin(y)} -\frac{sin(x)}{cos(x)}\)
Also, according to the basic identities:
\(\frac{cos(\theta)}{sin(\theta)} =cot(\theta)\\\\\frac{sin(\theta)}{cos(\theta)}= tan(\theta)\)
Therefore:
\(\frac{cos(x+y)}{cos(x)*sin(y)}=\frac{cos(y)}{sin(y)} -\frac{sin(x)}{cos(x)}=cot(y)-tan(x)\)
