Analysis Help! verify csc x/sinx-cotx/tanx=1?

Question

verify csc x/sinx-cotx/tanx=1

Answer 1

csc x/sinx - cotx/tanx = 1

Substitute for csc = 1/sin, cot = 1/tan...

(1/sinx)/sinx - (1/tanx)/tanx = 1
1/sin^2(x) - 1/tan^2(x) = 1

Since tan = sin/cos, 1/tan = cos/sin...

1/sin^2(x) - cos^2(x)/sin^2(x) = 1

Multiply by sin^2(x)...

1 - cos^2(x) = sin^2(x)

Move cos to the other side...

1 = sin^2(x) + cos^2(x)

... and this is a known identity.

Answer 2

csc x/sinx-cotx/tanx=1
Because csc x = 1/sin x and cot x = 1/ tan x the equation can be rewritten as follows:
csc x * cscx - cot x * cot x = 1
Multiplying csc's and cot' s together will result in the following:
csc^2 x - cot ^2 x = 1
One of the trig properties you learned is csc ^2 (x) = 1 + cot ^2 (x)
This problem is the same except cot^2 (x) is subtracted from both sides. Therefore:
1 = 1