Before looking through the proofs below, try to construct your own.
Note: The idea is always to drop perpendiculars, as right angle trigonometry comes in very handy. In most of these trigonometric formulas, in fact, it is very important to consider right angles.
1) cos(x+y)=cosx*cosy-sinx*siny
Let us construct a triangle ABC. We let D be the foot of perpendicular from A on BC. Then, let x=angle BAD and y=angle CAD. Also, let the length of AB=c, CD=e, AC=b, BD=d, AD=h. Below shows the diagram:
By cosine rule, we know that (d+e)^2=b^2+c^2-2bccos(x+y)
Also, by right angle trigonometry,
Triangle BAD: d=c*sinx, h=c*cosx
Triangle CAD: e=b*siny, h=b*cosy
Thus, we have (csinx)^2+2bc*sinx*siny+(bsiny)^2=b^2+c^2-2bc*cos(x+y)
Simplifying, we get 2bc(sinx*siny+cos(x+y))=(c*cosx)^2+(b*cosy)^2=2h^2
Thus, we have sinx*siny+cos(x+y)=cosx*cosy, and cos(x+y)=cosx*cosy-sinx*siny
2) sin(x+y)=sinx*cosy+cosx*siny
Using the same triangle, we consider areas.
Area of triangle ABC=(bc/2)*sin(x+y)=(d+e)(h/2)
Then, since h=c*cosx=b*cosy, d=csinx and e=bsiny, we have
h^2*sin(x+y)=(c*sinx+b*siny)*h*cosx*cosy=h^2(sinx*cosy+cosx*siny)
Hence, sin(x+y)=sinx*cosy+cosx*siny
From those identities, we can also write cos(x-y) and sin(x-y) in terms of cosx, cosy, sinx and siny. Try to find the final expressions.
Try the following exercises
1) Find the value of sin15 without using a calculator.
2) What is the value of sin90 and sin0? cos90 and cos0?
3) Show that cos(-x)=cosx and sin(-x)=-sinx
4) Remember the definition of a function? Plot a graph of f(x)=sinx and g(x)=cosx on the same axes. How did you extend the domain of the sin and cos functions?
5) Express tan(x+y) in terms of tanx and tany. Do the same for tan(x-y). Also, plot a graph of f(x)=tanx.
