Question

Prove tan (π- θ)= -tan θ

Answer 1

Not too sure what equations you have already seen to prove this.. But an easy way, if you've seen it, is by the difference for tan:
\[\tan(a-b)=\frac{\tan a-\tan b}{1+\tan a \tan b}\]
and the result becomes obvious.

Or, you could use the definition of tan in terms of sine and cosine, and use their difference formulas:
\[\tan(a-b)=\frac{\sin(a-b)}{\cos(a-b)}\]
and use the fact that
\(\sin(a-b)=\sin a \cos b-\sin b \cos a\)
\(\cos(a-b)=\cos a \cos b+\sin a \sin b\)

Or.. if you know these identities already: \(\sin(\pi-\theta)=\sin \theta\) and \(\cos(\pi-\theta)=-\cos \theta\), then you can use the definition of tan in terms of sine and cos and you get the result immediately.