We saw in the previous tutorial that

is true for every value of $\theta \in \mathbb{R}$

We also saw that

for any $\theta \in \mathbb{R}$ such that $\cos \theta \ne 0.$

Because these equations are true for every $\theta$ in their respective domains, they are called identities.

These identities are very useful, because they allow us to switch between $\sin ,\cos$ and $\tan ,$ as we will see in the next example.

1. Given that $\sin \theta =\frac{2}{5},$ and that $\theta$ is acute, find the values of $\cos \theta$ and $\tan \theta .$

2. $\phi$ is an obtuse angle satisfying ${\cos }^2\phi =\frac{3}{8}.$

   Find the value of $\tan \phi .$

Example ​

Prove the identities

1. $$6-3{\sin }^2\theta =3+3{\cos }^2\theta$$
2. $$\frac{1}{1-\cos \theta }=\frac{1+\cos \theta }{{\sin }^2\theta }$$
3. $${\cos }^2\theta +\sin \theta \cos \theta \tan \theta =1$$