Evaluate the expression![Screenshot 2025-05-28 123153.png](https://res.cloudinary.com/aptidude/image/upload/v1748415726/aptidude_questions/hgmhwazwzhw2fu9vpj4x.png)

Question

Accepted Answer

Step 1: Recall the Pythagorean identities:
1 + cot²θ = csc²θ
1 + tan²θ = sec²θ

Step 2: Substitute these into the expression:
(5) / (1 + cot²θ) + (3) / (1 + tan²θ) + 2cos²θ
= 5 / csc²θ + 3 / sec²θ + 2cos²θ

Step 3: Use reciprocal identities:
csc²θ = 1 / sin²θ => 5 / csc²θ = 5 * sin²θ
sec²θ = 1 / cos²θ => 3 / sec²θ = 3 * cos²θ

So the expression becomes:
5 sin²θ + 3 cos²θ + 2 cos²θ
= 5 sin²θ + 5 cos²θ

Step 4: Use sin²θ + cos²θ = 1:
5 (sin²θ + cos²θ) = 5 * 1 = 5

Therefore, the value of the expression is 5.

Answer

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