Prove that: (tan 69° + tan 66°)/(1 − tan 69° tan 66°) = −1

Question

Prove that:

\[ \frac{\tan 69^\circ+\tan 66^\circ} {1-\tan 69^\circ \tan 66^\circ} =-1 \]

Proof

Consider the left-hand side:

\[ \frac{\tan 69^\circ+\tan 66^\circ} {1-\tan 69^\circ \tan 66^\circ} \]

Using the identity:

\[ \tan(A+B) = \frac{\tan A+\tan B} {1-\tan A\tan B} \]

Let

\[ A=69^\circ, \qquad B=66^\circ \]

Then,

\[ = \tan(69^\circ+66^\circ) \]

\[ = \tan 135^\circ \]

We know that:

\[ \tan 135^\circ=-1 \]

Therefore,

\[ \frac{\tan 69^\circ+\tan 66^\circ} {1-\tan 69^\circ \tan 66^\circ} =-1 \]

Hence proved.