Given:
15 tan^2 A + 4 sec^2 A = 23

To find:
The value of (sec A cosec A)^2 - sin^2 A

Solution:

Step 1:
Let's simplify the given equation by replacing sec^2 A with (1 + tan^2 A) using the identity:
sec^2 A = 1 + tan^2 A

So, the equation becomes:
15 tan^2 A + 4(1 + tan^2 A) = 23

Step 2:
Simplify the equation further:
15 tan^2 A + 4 + 4 tan^2 A = 23

Combine like terms:
19 tan^2 A + 4 = 23

Step 3:
Subtract 4 from both sides:
19 tan^2 A = 19

Step 4:
Divide both sides by 19:
tan^2 A = 1

Step 5:
Taking the square root of both sides:
tan A = ±1

Step 6:
Since we have two possible values for tan A, let's consider each case separately:

Case 1: tan A = 1
In this case, we can substitute tan A = 1 into the equation (sec A cosec A)^2 - sin^2 A and simplify:

(sec A cosec A)^2 - sin^2 A
= (1/cos A * 1/sin A)^2 - sin^2 A
= (1/(cos A * sin A))^2 - sin^2 A
= (1/(cos A * sin A))^2 - (sin A)^2

Using the identity (sin A)^2 + (cos A)^2 = 1, we can rewrite the equation as:

= (1/(cos A * sin A))^2 - (1 - (cos A)^2)
= (1/(cos A * sin A))^2 - 1 + (cos A)^2
= (1/(cos A * sin A))^2 + (cos A)^2 - 1

Case 2: tan A = -1
In this case, we can substitute tan A = -1 into the equation (sec A cosec A)^2 - sin^2 A and simplify:

(sec A cosec A)^2 - sin^2 A
= (-1/cos A * -1/sin A)^2 - sin^2 A
= (1/(cos A * sin A))^2 - sin^2 A
= (1/(cos A * sin A))^2 - (sin A)^2

Using the identity (sin A)^2 + (cos A)^2 = 1, we can rewrite the equation as:

= (1/(cos A * sin A))^2 - (1 - (cos A)^2)
= (1/(cos A * sin A))^2 - 1 + (cos A)^2
= (1/(cos A * sin A))^2 + (cos A)^2 - 1

Step 7:
In both cases, we have the expression (1/(cos A * sin A))^2 + (cos A