What is the simplified value of (SecA/cotA+tanA)^2.

Solve this problem

We need to simplify the given expression:

(SecA / (cotA + tanA))²

Step 1: Express cotA and tanA in terms of sinA and cosA
We know that:
cotA = cosA / sinA
tanA = sinA / cosA

Substituting these values in the denominator:
cotA + tanA = (cosA / sinA) + (sinA / cosA)
= (cos²A + sin²A) / (sinA cosA)

Since sin²A + cos²A = 1, we get:
cotA + tanA = 1 / (sinA cosA)

Step 2: Express SecA
We know that secA = 1 / cosA. So, our given expression becomes:

(SecA / (cotA + tanA))²
= [(1 / cosA) / (1 / (sinA cosA))]²

Step 3: Simplify the fraction
Dividing by a fraction is the same as multiplying by its reciprocal:

(1 / cosA) × (sinA cosA / 1)
= sinA / cos²A

Now squaring the expression:

(sinA / cos²A)²
= sin²A / cos⁴A

Step 4: Final Answer
The simplified value of the given expression is sin²A / cos⁴A.