PROVE THAT (1+COT A+TAN A)(SIN A-COS A)=SEC A\COSEC A-COSEC A\SEC A

To prove the given equation (1 + cot A + tan A)(sin A - cos A) = sec A * cosec A - cosec A * sec A, we need to simplify both sides of the equation and show that they are equal.

Starting with the left-hand side (LHS):
(1 + cot A + tan A)(sin A - cos A)
First, notice that cot A is the reciprocal of tan A (cot A = 1/tan A), so we can rewrite it as:
(1 + 1/tan A + tan A)(sin A - cos A)

Next, use the property of reciprocals. The reciprocal of a sine function is cosec, and the reciprocal of a cosine function is sec. So we rewrite the equation as:
(1 + 1/tan A + tan A)((1/sin A) - (1/cos A))

Now, let's simplify the equation further.

1 + 1/tan A is equal to (tan A + 1)/tan A, so we can rewrite it as:
((tan A + 1)/tan A + tan A)((1/sin A) - (1/cos A))

Now, to simplify further, we need to find a common denominator for the fractions. The common denominator is sin A * cos A, so we can rewrite the equation as:
((tan A + 1)/tan A + tan A)((cos A/sin A) - (sin A/cos A))

Expanding the brackets:
((tan A + 1) + tan A^2)((cos A/sin A) - (sin A/cos A))

Since tan A^2 = sec^2 A - 1, we can replace tan A^2 with its equivalent:
((tan A + 1) + (sec^2 A - 1))((cos A/sin A) - (sin A/cos A))

Simplifying:
((tan A + 1 + sec^2 A - 1)((cos A/sin A) - (sin A/cos A))

The 1 and -1 terms cancel out:
(tan A + sec^2 A)((cos A/sin A) - (sin A/cos A))

Now, we need to simplify the right-hand side (RHS) of the equation:

sec A * cosec A - cosec A * sec A
Using the reciprocal property:
1/cos A * 1/sin A - 1/sin A * 1/cos A
1/(cos A * sin A) - 1/(cos A * sin A)
(1 - 1)/(cos A * sin A)
0/(cos A * sin A)
0

Therefore, the right-hand side simplifies to 0.

Comparing the simplified LHS and RHS:
(tan A + sec^2 A)((cos A/sin A) - (sin A/cos A)) = 0

We can see that the LHS does not equal the RHS, so the given equation (1 + cot A + tan A)(sin A - cos A) = sec A * cosec A - cosec A * sec A is not true.

Hence, the equation is not provable.