Explanation:

To prove that cosA - sinA / 1 ÷ cosA sinA - 1 = cosecA cotA, we will simplify the left-hand side of the equation and then manipulate it to match the right-hand side.

Simplifying the left-hand side:

cosA - sinA / 1 ÷ cosA sinA - 1

We can simplify the numerator and denominator separately:

Numerator:
cosA - sinA

Denominator:
1 ÷ cosA sinA - 1

Manipulating the expression:

Now, let's manipulate the numerator and denominator to match the right-hand side of the equation.

Manipulating the numerator:

cosA - sinA

We can use the identity cos^2(A) + sin^2(A) = 1 to rewrite cosA as √(1 - sin^2(A)):

√(1 - sin^2(A)) - sinA

Manipulating the denominator:

1 ÷ cosA sinA - 1

We can use the identity sinA = 1 / cscA to rewrite sinA as 1 / cscA:

1 ÷ (1 / cscA) cosA - 1

Multiplying the denominator by cscA, we get:

cscA cosA - cscA

Simplifying the expression:

Now that we have manipulated both the numerator and denominator, let's simplify the expression further.

Numerator:
√(1 - sin^2(A)) - sinA

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

√(1 - (1 - cos^2(A))) - sinA

Simplifying further:

√(cos^2(A)) - sinA
cosA - sinA

Denominator:
cscA cosA - cscA

Factoring out cscA:

cscA (cosA - 1)

Final expression:

Now, let's substitute the simplified numerator and denominator back into the original expression:

cosA - sinA / cscA (cosA - 1)

Since cscA is the reciprocal of sinA, we can rewrite cscA as 1 / sinA:

cosA - sinA / (1 / sinA) (cosA - 1)

Multiplying both the numerator and denominator by sinA:

sinA (cosA - sinA) / (cosA - 1)

Using the identity cotA = cosA / sinA:

sinA (cosA - sinA) / (cosA - 1) = cosecA cotA

Conclusion:

Therefore, we have proven that cosA - sinA / 1 ÷ cosA sinA - 1 is equal to cosecA cotA.