The expression tanA/(1−cotA) + cotA/(1−tanA) can be written as:

tanA/(1−cotA) + cotA/(1−tanA) = tanA/(1 - 1/tanA) + (1/tanA)/(1 - tanA)
= tan²A/(tanA - 1) - 1/(tanA(tanA - 1))
= (tan³A - 1)/(tanA(tanA - 1))
= (tanA - 1)(tan²A + tanA + 1)/(tanA(tanA - 1))
= (tan²A + tanA + 1)/tanA
= tanA + 1 + 1/tanA
= tanA + 1 + cotA
= tanA + cotA + 1

Alternatively,
Let the given expression be equal to X.
X = tanA/(1 - cotA) + cotA/(1 - tanA)
= tanA/(1 - 1/tanA) + (1/tanA)/(1 - tanA)
= tan²A/(tanA - 1) - 1/(tanA(tanA - 1))
= (tan³A - 1)/(tanA(tanA - 1))
= (tanA - 1)(tan²A + tanA + 1)/(tanA(tanA - 1))
= (tan²A + tanA + 1)/tanA
= tanA + 1 + cotA

Another approach:
X = tanA/(1 - 1/tanA) + 1/tanA/(1 - tanA)
= tan²A/(tanA - 1) - 1/(tanA(tanA - 1))
= (tan³A - 1)/(tanA(tanA - 1))
= (tanA - 1)(tan²A + tanA + 1)/tanA(tanA - 1)
= tan²A + tanA + 1/tanA
= tanA + cotA + 1
This is not among the given options. There must be a mistake in the question or the options.
Let's check another way:
Let's consider the expression:
X = tanA/(1 - cotA) + cotA/(1 - tanA)
= sinA/cosA / (1 - cosA/sinA) + cosA/sinA / (1 - sinA/cosA)
= sin²A/cosA(sinA - cosA) + cos²A/sinA(cosA - sinA)
= (sin³A - cos³A)/cosA sinA (sinA - cosA)
= (sinA - cosA)(sin²A + sinAcosA + cos²A)/cosA sinA (sinA - cosA)
= (1 + sinAcosA)/cosA sinA
= 1/sinAcosA + 1 = secAcosecA + 1