Given: sec A = 2/√3

To find: (tan A / cos A) * (1 + sin A / tan A)

Approach:
1. Recall the definitions of sec A, tan A, and sin A.
2. Substitute the given value of sec A into the expression.
3. Simplify the expression using trigonometric identities.
4. Evaluate the final expression.

Solution:

Step 1: Recall the definitions of sec A, tan A, and sin A.
- sec A = 1 / cos A
- tan A = sin A / cos A
- sin A = tan A * cos A

Step 2: Substitute the given value of sec A into the expression.
- sec A = 2/√3

Step 3: Simplify the expression using trigonometric identities.
We can rewrite sec A as 1/cos A:
- 1/cos A = 2/√3

To simplify further, we can cross-multiply:
- √3 = 2 * cos A

Divide both sides by 2:
- cos A = √3 / 2

Now, substitute the value of cos A into the expression for sin A:
- sin A = tan A * cos A
- sin A = (sin A / cos A) * (cos A)

Cancel out the cos A terms:
- sin A = sin A

Step 4: Evaluate the final expression.
Now, substitute the simplified values back into the given expression:
- (tan A / cos A) * (1 + sin A / tan A)
- [(sin A / cos A) / cos A] * (1 + sin A / (sin A / cos A))
- [(sin A / cos A) / cos A] * (1 + cos A)
- (sin A / cos A) * (1 + cos A)
- sin A + sin A * cos A / cos A
- sin A + sin A
- 2 * sin A

Since sin A = sin A, we can simplify further:
- 2 * sin A = 2

Therefore, the value of (tan A / cos A) * (1 + sin A / tan A) is 2.