Given cosecA + cotA = 3, evaluate cosecA - cotA and cosA?

Question

Please, I have tried for a long time yet I can't figure out how to solve this!

Answer 1

csc A + cot A = 3

Use the trig identity:
1 + cot^2 A = csc^2 A

Rewrite it as:
csc^2 A - cot^2 A = 1

Remember difference of 2 squares? It makes this easy:
csc^2 A - cot^2 A = (csc A + cot A) (csc A - cot A)

Substitute:
1 = 3 * (csc A - cot A)
(csc A - cot A) = 1/3 (solution!)

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To find cos A:
Use the identities for csc and cot:
csc A = 1/sin A
cot A = cos A / sin A

Substitute into csc A + cot A and csc A - cot A:
(csc A - cot A) = 1/sin A - cos A / sin A = (1 - cos A)/sin A = 1/3
1 - cos A = 1/3 sin A
cos A = 1 - 1/3 sin A

(csc A + cot A) = 1/sin A + cos A / sin A = (1 + cos A)/sin A = 3
(1 + cos A) = 3 sin A
cos A = 3 sin A - 1

Set the 2 values of cos A equal to each other:
1 - 1/3 sin A = 3 sin A - 1
2 = 3 1/3 sin A
3/5 = sin A

Substitute our value of sin A into cos A:
cos A = 3 sin A - 1
cos A = 3 (3/5) - 1
cos A = 4/5 (solution)

Check with the other value of cos A:
cos A = 1 - 1/3 sin A
cos A = 1 - 1/3 (3/5)
cos A = 1 - 1/5 = 4/5 (check)