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If SinA = 3/5 and CosB = 5/13, then find the value of Cot(A + B).


1. 16/36
2. 16/53
3. -16/63
4. 16/63

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Correct Answer - Option 3 : -16/63

Given:

SinA = 3/5

CosB = 5/13

Formula:

Cot(A + B) = (CotA.CotB – 1)/(CotB + CotA)

Concept Used:

3, 4 and 5 are Pythagorean triplets.

5, 12 and 13 are Pythagorean triplets.

Calculation:

We know 3, 4 and 5 are Pythagorean triplets.

If SinA = 3/5 then CosA = 4/5

CotA = CosA/SinA

⇒ (4/5)/(3/5) = 4/3

Again 5, 12, 13 are Pythagorean triplets.

If CosB = 5/13 then SinB = 12/13

CotB = CosB/SinB 

⇒ (5/13)/(12/13) = 5/12

Now Cot(A + B) = (CotA CotB – 1)/(CotB + CotA)

⇒ Cot(A + B) = {(4/3) × (5/12) – 1)}/(5/12  + 4/3)

⇒ Cot(A + B) = {(20/36) – 1}/(21/12)

⇒ Cot(A + B) = {(5/9) – 1}/(7/4)

⇒ Cot(A + B) = (-4/9)/(7/4)

⇒ Cot(A + B) = (-16)/(63)       

The value of Cot(A + B) = -16/63

Some important formula to remember:-

1) Sin(A + B) = SinA CosB + CosA SinB

2) Sin(A - B) = SinA CosB - CosA SinB

3) Cos(A + B) = CosA CosB - SinA SinB

4) Cos(A - B) = CosA CosB + SinA SinB

5) Tan(A + B) = (TanA + Tan B)/(1 - TanA TanB)

6) Tan(A - B) = (TanA - TanB)/(1 + TanA TanB)

7) Cot(A + B) = (CotA CotB - 1)/(CotB + CotA)

8) Cot(A - B) = (CotA CotB + 1)/(CotB - CotA)

Some important Triplets to remember:-

1) 3, 4, 5

2) 5, 12, 13

3) 7, 24, 25

4) 8, 15, 17

5) 9, 40, 41

 Be careful while marking the answer. Option 3 and 4 have the same value but we have to mark 16/63 with the negative sign.

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