NEED ASAP! verify the following identity: (see below)

Question

cos(α - β)

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ = cot α + cot β

sin α cos β

(HINT: if you start with the left hand side, there are roughly 5 steps)

please show every step

Answer 1

I believe you have a typo in your problem because the left side does not equal the right side.

Use the sum and difference identity:

to change the left side into:

[cos(α)cos(β) + sin(α)sin(β)]/sin(α)cos(β)

Then split the fraction into two parts (essentially go backwards from a single numerator divided by the common denominator into two separate fractions both having the common denominator):

[cos(α)cos(β)]/[sin(α)cos(β)] + [sin(α)sin(β)]/[sin(α)cos(β)]

Now cancel cos(β) from the top and bottom of the left fraction and cancel sin(α) from the top and bottom of the right fraction leaving:

cos(α)/sin(α) + sin(β)/cos(β)

Then substitute cos(α)/sin(α) = cot(α) and sin(β)/cos(β) = tan(β) leaving:

cot(α) + tan(β)