What is the value of tan A + cot A if tan A = 2?
- (a)5⁄2
- (b)3⁄2
- (c)1⁄2
- (d)7⁄2
Answer
A
Why
Correct — A.
cot A = 1 ⁄ tan A = 1⁄2
tan A + cot A = 2 + 1⁄2
= 4⁄2 + 1⁄2 = 5⁄2 → option (a)
Why the others are wrong
- (b)3⁄2 — 3⁄2 is tan A − cot A = 2 − 1⁄2. The question adds the two, so the answer is 2 + 1⁄2 = 5⁄2.
- (c)1⁄2 — 1⁄2 is cot A alone. The question asks for tan A + cot A, so tan A = 2 still has to be added.
- (d)7⁄2 — 7⁄2 would need cot A = 3⁄2. But cot A is the reciprocal of tan A, so cot A = 1⁄2 and the sum is 5⁄2.
Concept
cot A is the reciprocal of tan A: cot A = cos A ⁄ sin A = 1 ⁄ tan A. So tan A + cot A = t + 1⁄t, where t = tan A.
The sum also equals 1 ⁄ (sin A cos A), because (sin²A + cos²A) ⁄ (sin A cos A) = 1 ⁄ (sin A cos A). Here that makes sin A cos A = 2⁄5.
No quadrant is given, and none is needed: cot A = 1 ⁄ tan A holds wherever tan A = 2, so the sum is 5⁄2 either way.
Key facts
- cot A = 1 ⁄ tan A.
- tan A + cot A = 1 ⁄ (sin A cos A) = sec A × cosec A.
- For t > 0, t + 1⁄t ≥ 2, with equality at t = 1, that is A = 45°.
Study next
tan A + cot A = 2 for an acute angle, which forces A = 45°tan²A + cot²A from tan A + cot A: square the sum and subtract 2sin A cos A and sec A × cosec A from a given tan A
Common traps
- Stopping at cot A = 1⁄2 and forgetting to add tan A = 2.
- Subtracting the two, 2 − 1⁄2 = 3⁄2, when the question adds them.
The reverse runs at 19 Sep 2024, 16:00, Quant Q.17: tan θ + cot θ = 2 for an acute angle gives θ = 45° (option c).
19 Sep 2025, 09:00, Quant Q.14 builds on it: tan A + cot A = 2 gives sin A + cos A = √2 (option b).
Related PYQs
No directly related past PYQ was found.