Given, cosec²A – cot²A = 1

cosec²A = 1 + cot²A

Since cosec function is the inverse of sin function, it is written as

1/sin²A = 1 + cot²A

Now, rearrange the terms, it becomes

sin²A = 1/(1+cot²A)

Now, take square roots on both sides, we get

sin A = ±1/(√(1+cot²A)

The above equation defines the sin function in terms of cot function

sin²A = 1/ (1+cot²A)

1 – cos²A = 1/ (1+cot²A)

Rearrange the terms,

cos²A = 1 – 1/(1+cot²A)

⇒ cos²A = (1-1+cot²A)/(1+cot²A)

⇒ 1/sec²A = cot²A/(1+cot²A)

⇒ sec A = ±√ (1+cot²A)/cotA

Now, to express tan function in terms of cot function

tan A = sin A/cos A and cot A = cos A/sin A

Since cot function is the inverse of tan function, it is rewritten as

tan A = 1/cot A