To express trigonometric ratios sin A, sec A, and tan A in terms of cot A, we need to use the reciprocal identities and the Pythagorean identity. Here's a step-by-step explanation:

Reciprocal Identities:
1. The reciprocal of sin A is cosec A.
- cosec A = 1 / sin A

2. The reciprocal of cos A is sec A.
- sec A = 1 / cos A

3. The reciprocal of tan A is cot A.
- cot A = 1 / tan A

Pythagorean Identity:
The Pythagorean identity states that for any angle A in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
- a^2 + b^2 = c^2

Deriving sin A in terms of cot A:
1. Start with the Pythagorean identity:
- a^2 + b^2 = c^2

2. Divide both sides by c^2:
- (a^2 / c^2) + (b^2 / c^2) = 1

3. Recall that sin A = b / c and cos A = a / c:
- (sin A)^2 + (cos A)^2 = 1

4. Rearrange the equation:
- (sin A)^2 = 1 - (cos A)^2

5. Substitute 1 - (cos A)^2 with (1 / (cos A)^2) * (cos A)^2:
- (sin A)^2 = (1 / (cos A)^2) * (cos A)^2 - (cos A)^2

6. Factor out (cos A)^2:
- (sin A)^2 = [(1 / (cos A)^2) - 1] * (cos A)^2

7. Simplify the expression inside the brackets:
- (sin A)^2 = [(1 - (cos A)^2) / (cos A)^2] * (cos A)^2

8. Recall that cot A = cos A / sin A:
- (sin A)^2 = [(1 - (cos A)^2) / (cos A)^2] * (1 / cot A)

9. Take the square root of both sides to solve for sin A:
- sin A = sqrt[(1 - (cos A)^2) / (cos A)^2] * (1 / cot A)

Deriving sec A in terms of cot A:
1. Start with the reciprocal identity for sec A:
- sec A = 1 / cos A

2. Multiply the numerator and denominator by sin A:
- sec A = (1 / cos A) * (sin A / sin A)

3. Recall that cot A = cos A / sin A:
- sec A = (cot A * sin A) / sin A

4. Cancel out sin A in the denominator:
- sec A = cot A

Deriving tan A in terms of cot A:
1. Start with the reciprocal identity for tan A:
- tan A = 1 / cot A

1)
sinA= 1/cosecA = 1 / √(1+cot²A)

[ cot²A+ 1 = cosec²A,
cosecA= √( 1+cot²A)]

2)
tanA= 1/cotA

3)
secA= √(1+tan²A)

[sec²A= 1+tan²A , secA= √ (1+tan²A)]

secA= √(1+ (1/cot²A)) = √ (1+1/ cot²A)
secA = √(cot²A+1/cot²A)

secA= √1+cot²A/ (cotA)