if 0<A<90 and

Sin A+ cos A+ tan A+cot A+sec A+cosec A =7

then prove that sin 2A is a root of (x)^2 -44x-36=0

To prove that sin 2A is a root of the equation (x)^2 - 44x - 36 = 0, we need to use trigonometric identities.

Let's start by expressing sin 2A in terms of A using the double-angle formula for sine.

sin 2A = 2 sin A cos A

Now, let's rewrite the given expression Sin A+ cos A+ tan A+cot A+sec A+cosec A = 7 using trigonometric identities.

sin A + cos A + tan A + cot A + sec A + cosec A
= (sin A / cos A) + cos A + (sin A / cos A) + (cos A / sin A) + (1 / cos A) + (1 / sin A)

To combine the terms, let's find a common denominator:

= (sin^2 A + cos^2 A) / (sin A cos A) + (cos^2 A + sin^2 A) / (cos A sin A) + (sin A + cos A) / (cos A sin A)

Since sin^2 A + cos^2 A = 1, and sin A / cos A = 1 / (cos A / sin A), we can simplify the expression:

= 1 / (cos A sin A) + 1 / (cos A sin A) + (sin A + cos A) / (cos A sin A)
= 2 / (cos A sin A) + (sin A + cos A) / (cos A sin A)
= (2 + sin A + cos A) / (cos A sin A)

Now, we have (2 + sin A + cos A) / (cos A sin A) = 7. Let's solve for sin A and cos A separately using this equation:

2 + sin A + cos A = 7 cos A sin A

Rearranging the terms:

7 cos A sin A - sin A - cos A + 2 = 0

Factoring out sin A and cos A:

(sin A - 1)(7 cos A - 2) = 0

Now, we have two possible cases:

Case 1: sin A - 1 = 0
This implies sin A = 1, which is not possible when 0 < A < 90 degrees.

Case 2: 7 cos A - 2 = 0
This implies cos A = 2/7.

Now, let's find sin 2A using the double-angle formula:

sin 2A = 2 sin A cos A
= 2 sin A (2/7)
= 4 sin A / 7

Since cos A = 2/7, we can substitute this into the expression:

sin 2A = 4 (2/7) / 7
= 8/49

Thus, sin 2A = 8/49.

Now, let's check if sin 2A is a root of the equation (x)^2 - 44x - 36 = 0:

Substituting sin 2A = 8/49 into the equation:

(8/49)^2 - 44(8/49) - 36 = 0
64/2401 - 352/49 - 1764/49 = 0
64 - 352 - 1764 = 0

Simplifying, we get:

-2052 = 0

Since -2052 is not equal to 0, we conclude that sin 2A is not a root of the equation (x)^2 - 44x - 36 = 0.

Therefore, the statement is not true.