Proof:

We need to prove that (SinA cosecA)^2 (cosA secA)^2 = 7 tan^2A cot^2A.

Step 1: Rewrite the left side of the equation using trigonometric identities

Using the reciprocal identities, we can rewrite the left side of the equation as:

(SinA cosecA)^2 (cosA secA)^2 = ((SinA/SinA)(1/SinA))^2 * ((cosA/cosA)(1/cosA))^2

Simplifying this expression, we get:

(1/SinA)^2 * (1/cosA)^2 = (1/Sin^2A) * (1/cos^2A)

Step 2: Rewrite the right side of the equation using trigonometric identities

Using the identities tanA = SinA/cosA and cotA = cosA/SinA, we can rewrite the right side of the equation as:

7 tan^2A cot^2A = 7 (SinA/cosA)^2 * (cosA/SinA)^2

Simplifying this expression, we get:

7 (Sin^2A/cos^2A) * (cos^2A/Sin^2A) = 7 (Sin^2A * cos^2A) / (cos^2A * Sin^2A)

Step 3: Cancel out common factors

As we can see, the expression on the left side and the expression on the right side are equal. Additionally, we can observe that (Sin^2A * cos^2A) and (cos^2A * Sin^2A) are the same, as they are multiplication of the same factors. Therefore, we can cancel out these common factors from the numerator and denominator:

(1/Sin^2A) * (1/cos^2A) = 7

Step 4: Simplify the expression

To simplify further, we can multiply both sides of the equation by (Sin^2A * cos^2A):

1 = 7 (Sin^2A * cos^2A)

Now, we can divide both sides of the equation by 7:

1/7 = Sin^2A * cos^2A

Step 5: Apply the Pythagorean identity

Using the Pythagorean identity, Sin^2A + cos^2A = 1, we can substitute this expression into our equation:

1/7 = (1 - cos^2A) * cos^2A

Expanding the equation, we get:

1/7 = cos^2A - cos^4A

Step 6: Rearrange the equation

Rearranging the equation, we have:

cos^4A - cos^2A + 1/7 = 0

Step 7: Solve the equation

This quadratic equation can be solved using factoring or the quadratic formula. However, upon closer inspection, we can see that this equation does not have real solutions. Therefore, we conclude that the equation is not valid.

Therefore, we cannot prove that (SinA cosecA)^2 (cosA secA)^2