Trigonometric Identities

Video lesson · from 44:071 worked example

One lesson video covers all of Trigonometry, so it opens at 44:07 for this topic — not from the beginning.

You will often be asked to prove trigonometric identities by showing that one side equals the other.

Useful tools for doing this include:

Work on one side only: start with the more complicated side and transform it until it matches the other. Moving terms across the equals sign assumes what you are trying to prove.
$\sin^2 x + \cos^2 x = 1$ is the workhorse: and its rearrangements $\sin^2 x = 1 - \cos^2 x$ and $\cos^2 x = 1 - \sin^2 x$ are what let you switch between functions.
Convert tangent early: replacing $\tan x$ with $\frac{\sin x}{\cos x}$ usually unlocks the problem.
Finish the argument: end by stating that the two sides are now equal. An unfinished chain of algebra does not read as a proof.

Example 1

Show that $\frac{\sin(s + t)}{\cos s \cos t} = \tan s + \tan t$ for $\cos s \neq 0$ and $\cos t \neq 0$ .

Start with the LHS and expand the numerator using the addition formula:

LHS = \frac{\sin s \cos t + \cos s \sin t}{\cos s \cos t}

Split the fraction into two parts over the common denominator:

LHS = \frac{\sin s \cos t}{\cos s \cos t} + \frac{\cos s \sin t}{\cos s \cos t}

Cancel common terms in each fraction:

LHS = \frac{\sin s}{\cos s} + \frac{\sin t}{\cos t}

Since $\tan x = \frac{\sin x}{\cos x}$ :

LHS = \tan s + \tan t = RHS

As completely shown.