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.
