Correct Answer - Option 2 :
\(\dfrac{\pi}{4}\)
Concept:
- tan (A ± B) = \(\rm \dfrac{\tan A ± \tan B}{1 ∓ \tan A \tan B}\).
-
\(\rm \tan\dfrac{\pi}{4}=1\).
Calculation:
It is given that \(\rm \tan \alpha = \dfrac{m}{m+1}\) and \(\rm \tan \beta = \dfrac{1}{2m+1}\).
∴ tan (α + β)
= \(\rm \dfrac{\tan \alpha +\tan \beta}{1 -\tan \alpha \tan \beta}\)
= \(\rm \dfrac{\tfrac{m}{m+1} +\tfrac{1}{2m+1}}{1 -\left (\tfrac{m}{m+1} \right )\left (\tfrac{1}{2m+1} \right )}\)
= \(\rm \dfrac{m(2m+1)+(m+1)}{(m+1)(2m+1)-m}\)
= \(\rm \dfrac{2m^2+m+m+1}{2m^2+m+2m+1-m}\)
= 1
= \(\rm \tan\left(n\pi+\dfrac{\pi}{4}\right)\), n ∈ Z.
∴ α + β = \(\rm \dfrac{\pi}{4}\).
- sin (A ± B) = sin A cos B ± sin B cos A.
- cos (A ± B) = cos A cos B ∓ sin A sin B.
