Prove that (1 + cotA - cosecA)(1 + tanA + secA)=2

We are given the following expression to prove:

(1 + cotA - cosecA)(1 + tanA + secA) = 2

Step 1: Write the trigonometric identities in terms of sine and cosine.

We know the following trigonometric identities:

cotA = cosA/sinA cosecA = 1/sinA tanA = sinA/cosA secA = 1/cosA

Now, substitute these identities into the given expression:

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

Step 2: Simplify each part.

Simplify the first part of the expression:

(1 + cosA/sinA - 1/sinA) = (1 - 1/sinA) + cosA/sinA = (sinA - 1 + cosA)/sinA

Now simplify the second part:

(1 + sinA/cosA + 1/cosA) = (cosA + sinA + 1)/cosA

Step 3: Multiply the two parts.

Now, multiply the simplified expressions:

[(sinA - 1 + cosA)/sinA] × [(cosA + sinA + 1)/cosA]

This simplifies to:

[(sinA - 1 + cosA)(cosA + sinA + 1)] / (sinA × cosA)

Step 4: Expand the numerator.

Now, expand the numerator:

(sinA - 1 + cosA)(cosA + sinA + 1)

= sinA(cosA + sinA + 1) - 1(cosA + sinA + 1) + cosA(cosA + sinA + 1)

Expanding each term:

= sinA cosA + sinA^2 + sinA - cosA - sinA - 1 + cosA^2 + cosA sinA + cosA

= sinA cosA + sinA^2 + sinA - cosA - sinA - 1 + cosA^2 + cosA sinA + cosA

Now, combine like terms:

= (sinA cosA + cosA sinA) + (sinA^2 + cosA^2) + (sinA - sinA - cosA + cosA) - 1

= 2 sinA cosA + (sinA^2 + cosA^2) - 1

Since sinA^2 + cosA^2 = 1, we have:

= 2 sinA cosA + 1 - 1

= 2 sinA cosA

Step 5: Final simplification.

Now, substitute the expanded numerator back into the expression:

(2 sinA cosA) / (sinA cosA)

This simplifies to:

2

Therefore, we have proven that:

(1 + cotA - cosecA)(1 + tanA + secA) = 2.