Trigonometric Ratio & Identities: If alpha+beta+gamma=2pi , then | IIT-JEE 1979

IIT-JEE 1979Question 1

If $\alpha+\beta+\gamma=2\pi$, then

Trigonometric Ratio & Identities

$\tan\frac{\alpha}{2}+\tan\frac{\beta}{2}+\tan\frac{\gamma}{2}=\tan\frac{\alpha}{2}\tan\frac{\beta}{2}\tan\frac{\gamma}{2}$✓ Correct answer$\tan\frac{\alpha}{2}\tan\frac{\beta}{2}+\tan\frac{\beta}{2}\tan\frac{\gamma}{2}+\tan\frac{\gamma}{2}\tan\frac{\alpha}{2}=1$$\tan\frac{\alpha}{2}+\tan\frac{\beta}{2}+\tan\frac{\gamma}{2}=-\tan\frac{\alpha}{2}\tan\frac{\beta}{2}\tan\frac{\gamma}{2}$None of these

Solution

Concept. $\frac\alpha2+\frac\beta2+\frac\gamma2=\pi$, so $\tan\left(\frac\alpha2+\frac\beta2\right)=-\tan\frac\gamma2$. Formulas used. (1) $\tan(X+Y)=\dfrac{\tan X+\tan Y}{1-\tan X\tan Y}$Step 1 - Use (1). With $x,y,z$ the three half-angle tangents: $$\frac{x+y}{1-xy}=-z\ \Longrightarrow\ x+y=-z+xyz.$$ Step 2 - Rearrange. $x+y+z=xyz$, i.e. option (A). Answer: (A)

Trigonometric Ratio & Identities - JEE Mathematics question (IIT-JEE 1979)

More from Trigonometric Ratio & Identities