- #1

opus

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## Main Question or Discussion Point

I'll start off with a given problem.

Find ##cos\left(α-β\right)## given that

##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III.

##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.

Solution:

##cos\left(α-β\right)=1##

This had we wonder if this continued for other angles of the same measure so I tried different inputs such as ##cos\left(30°-30°\right)## and again the solution was 1 after using the difference of cosines formula.

My question is, when we take the difference of cosines with the same angle, is it always equal to 1?

And following this train of thought, could it be true that in taking the difference of sines with the same angle, this would be equal to 0?

And would this extend to addition of the angles as well?

Find ##cos\left(α-β\right)## given that

##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III.

##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.

Solution:

##cos\left(α-β\right)=1##

This had we wonder if this continued for other angles of the same measure so I tried different inputs such as ##cos\left(30°-30°\right)## and again the solution was 1 after using the difference of cosines formula.

My question is, when we take the difference of cosines with the same angle, is it always equal to 1?

And following this train of thought, could it be true that in taking the difference of sines with the same angle, this would be equal to 0?

And would this extend to addition of the angles as well?