\(\displaystyle \begin&=\frac=\frac=\frac\\&=\frac\cdot \frac\\&=\frac\\&=\frac\\\frac&=\frac\,\,\,\,\,\,\,\,\surd \end\) Note: The right-hand side looked a little more complicated (because of the tangents) so we started there.

We turned the tangents into sines and cosines and simplified first.

We now proceed to derive two other related formulas that can be used when proving trigonometric identities.

It is suggested that you remember how to find the identities, rather than try to memorise each one.

\(\displaystyle \begin&=\frac-\frac\\\tan \left( \right)&=\frac\,\,\,\,\,\surd \end\) Note: Start from the right side, and turn everything into sin and cos since the Half Angle tan identity is written in terms of sin and cos.

\(\displaystyle \begin\frac&=\\frac&=\cos \theta \sin \theta \,\,\,\,\,\surd \end\) Note: We knew to use \(\displaystyle \theta -\theta \) and difference of squares for \(\cos 2\theta \) since the denominator contains both cos and sin.

For example, for \(\displaystyle \sin \left( \right)\), if \(\displaystyle \frac\) is in the first or second quadrants, use \(\displaystyle \sqrt\), and if \(\displaystyle \frac\) is in the third or fourth quadrants, use \(\displaystyle -\sqrt\).

Similarly for \(\displaystyle \cos \left( \right)\), if \(\displaystyle \frac\) is in the first or fourth quadrants, use \(\displaystyle \sqrt\), and if \(\displaystyle \frac\) is in the second or third quadrants, use \(\displaystyle -\sqrt\).

Note how we work on one side only and pull down the other side when it matches.

It doesn’t matter which side we start on, but typically, it’s the most complicated.

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