Let's prove this supplementary angle identities using the cosine and sine addition formulas. The supplementary angle identities are the ones that say sine of pi minus theta equals blank or cosine of pi minus theta equals blank.
Pi minus theta is the supplement of theta. Well the sine of a difference is sine, cosine, cosine, sine. Sine pi, cosine theta, cosine pi, sine theta and with sine the minus stays the same. The sine of pi 0, cosine of pi is -1, so it turns out that the sine of pi minus theta is equal to the sine of theta. That's a really important in resolving you use a lot when you're solving trigonometric equation involving sine.
Supplementary angles have the same output like 30 degrees and 150 degrees would have the same output, like 30 degrees and 150 degrees would have the same output, what about cosine is this true? Well for cosine of a difference we have cosine, cosine, sine, sine that's cosine pi, cosine theta minus sine pi, sine theta. Again cosine pi is -1, sine of pi is 0, so we're left with minus cosine theta it's not true. If you take the supplement of an angle, its cosine will be opposite.