# Relation between Trigonometric Functions of Same Argument

We already know that cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta).

If we take alpha=t,beta=t then cos(t-t)=cos(t)cos(t)+sin(t)sin(t) or cos(0)=cos^2(t)+sin^2(t).

This gives us very important identity that connects sine and cosine (it is also called main trigonometric identity): cos^2(t)+sin^2(t)=1.

Dividing main trigonometric identity by cos^2(t) gives that 1+tan^2(t)=1/(cos^2(t))=sec^2(t). Dividing main trigonometric identity by sin^2(t) gives that 1+cot^2(t)=1/(sin^2(t))=csc^2(t).

So, we have three formulas:

1. color(red)(cos^2(t)+sin^2(t)=1),
2. color(blue)(1+tan^2(t)=sec^2(t)),
3. color(green)(1+cot^2(t)=csc^2(t)).

Formula (2) holds when t!=pi/2+pin, n in ZZ (when cos(t)!=0).Formula (3) holds when t!=pin, n in ZZ (when sin(t)!=0).

These three formulas connect different trigonometric functions of same argument.

There are also two additional identities that connect different trigonometric functions of same argument: tan(t)=(sin(t))/(cos(t)),cot(t)=(cos(t))/(sin(t)) . But in fact these identities are definitions of tangent and cotangent.

Multiplying above two identities, we will obtain relation tan(t)xxcot(t)=1 that is true when t!=(pi k)/2, k in ZZ (when both cos(t)!=0 and sin(t)!=0).

Example. It is known that sin(t)=-3/5 and pi<t<(3pi)/2. Find cos(t),tan(t),cot(t).

From formula (1) we have that cos^2(t)=1-sin^2(t)=1-(-3/5)^2=1-9/25=16/25.

Since cos^2(t)=16/25, then either cos(t)=sqrt(16/25)=4/5 or cos(t)=-sqrt(16/25)=-4/5.

Since pi<t<(3pi)/2, then t belongs to third quadrant. In this quadrant cosine is negative, so cos(t)=-4/5.

Now, tan(t)=(sin(t))/(cos(t))=(-3/5)/(-4/5)=3/4 and cot(t)=1/(tan(t))=4/3.

So, cos(t)=-4/5,tan(t)=3/4,cot(t)=4/3.