Question

Verify the basic identity. What is the domain of validity? cot theta = cos theta csc theta

Answer 1

So far I got :
cotx=cosx(1/sinx)
cotx=cosx/sinx
cotx=cotx

Is that right for verifying the basic identity? And how do I find the domain of validity?

Answer 2

for which values of x the function is defined

Answer 3

Domain of validity excludes division by zero, which would be where opposite side length is zero, meaning 0 degrees or 180 degrees. So domain of validity is all angles except multiples of 180 degrees (multiples of pi, in radians).

Answer 4

cot x = cos x / sin x right , the values of x for which sinx = 0 will not be a valid domain

Answer 5

So how do I find those values?

Answer 6

It's all angles except 0, 180, 360, 540, ... (degrees).

Answer 7

More precisely, the domain of validity is all angles except those in {0, +/-pi, +/-2pi, +/- 3pi, ...} (radians) or {0, +/-180, +/-360, ...} (degrees).

Answer 8

Wait, so i'm a little confused, what kind of work can I show on this problem (i'm supposed to show my work)

Answer 9

just draw the sine curve and mark the points where sinx = 0 and say those values are excluded

Answer 10

I can't do drawings^

Answer 11

Cotangent is A/O. Cosine is A/H. Cosecant is H/O. To prevent division by zero, O must not be zero and H must not be zero. H is never zero, but O is zero at angles of 0, +/-pi, +/-2pi, etc. So the domain of validity = {x : x not = 0, +/-pi, +/-2pi, ...}. That's the complete solution.