Question

Prove: cos(x+y)/cosxsiny=coty-tanx

Answer 1

cos (x+y)/cos x sin y    

= cos x cos y - sin x sin y  / cos x sin y

= cos x cos y/cos x sin y     - sin x sin y / cos x sin y

hope this helps

Answer 2

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:

$\frac{a \pm b}{c} =\frac{a}{c} \pm \frac{b}{c} \hspace{3} c\neq0$

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:

$\frac{cos(x+y)}{cos(x)*sin(y)}=\frac{cos(y)}{sin(y)} -\frac{sin(x)}{cos(x)}=cot(y)-tan(x)$