If x: y: z = tan left(frac{pi}{15}+alpharight | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given that $x: y: z = 	an \left(12^{\circ}+\alphaight): 	an \left(12^{\circ}+\betaight): 	an \left(12^{\circ}+\gammaight)$.Let $x = k 	a

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Given that $x: y: z = 	an \left(12^{\circ}+\alphaight): 	an \left(12^{\circ}+\betaight): 	an \left(12^{\circ}+\gammaight)$.Let $x = k 	an \left(12^{\circ}+\alphaight)$,$y = k 	an \left(12^{\circ}+\betaight)$,and $z = k 	an \left(12^{\circ}+\gammaight)$.Consider the term $\frac{z+x}{z-x} \sin ^2(\gamma-\alpha)$.Using the formula $\frac{	an A + 	an B}{	an A - 	an B} = \frac{\sin(A+B)}{\sin(A-B)}$,we get:$\frac{z+x}{z-x} = \frac{	an(12^{\circ}+\gamma) + 	an(12^{\circ}+\alpha)}{	an(12^{\circ}+\gamma) - 	an(12^{\circ}+\alpha)} = \frac{\sin(24^{\circ} + \gamma + \alpha)}{\sin(\gamma - \alpha)}$.Thus,$\frac{z+x}{z-x} \sin ^2(\gamma-\alpha) = \sin(24^{\circ} + \gamma + \alpha) \sin(\gamma - \alpha)$.Using $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$,we have:$= \frac{1}{2} [\cos(24^{\circ} + \gamma + \alpha - \gamma + \alpha) - \cos(24^{\circ} + \gamma + \alpha + \gamma - \alpha)]$$= \frac{1}{2} [\cos(24^{\circ} + 2\alpha) - \cos(24^{\circ} + 2\gamma)]$.Similarly,$\frac{x+y}{x-y} \sin ^2(\alpha-\beta) = \frac{1}{2} [\cos(24^{\circ} + 2\beta) - \cos(24^{\circ} + 2\alpha)]$ and $\frac{y+z}{y-z} \sin ^2(\beta-\gamma) = \frac{1}{2} [\cos(24^{\circ} + 2\gamma) - \cos(24^{\circ} + 2\beta)]$.Adding these three expressions,we get:$\frac{1}{2} [\cos(24^{\circ} + 2\alpha) - \cos(24^{\circ} + 2\gamma) + \cos(24^{\circ} + 2\beta) - \cos(24^{\circ} + 2\alpha) + \cos(24^{\circ} + 2\gamma) - \cos(24^{\circ} + 2\beta)] = 0$.

If $x: y: z = \tan \left(\frac{\pi}{15}+\alpha\right): \tan \left(\frac{\pi}{15}+\beta\right): \tan \left(\frac{\pi}{15}+\gamma\right)$,then find the value of $\frac{z+x}{z-x} \sin ^2(\gamma-\alpha)+\frac{x+y}{x-y} \sin ^2(\alpha-\beta)+\frac{y+z}{y-z} \sin ^2(\beta-\gamma)$.

Similar Questions

If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=p$,then $\tan p$ is

If $1-\cot 23^{\circ}=\frac{x}{1-\cot 22^{\circ}}$,then $x=$

For $n \in N$,if $f(n) = (\cos nx)(\sec x)^n$ and $g(n) = (\sin nx)(\sec x)^n$,then $f(2020) - f(2019) + (\tan x)g(2019) =$

If $\theta$ is eliminated from the equations $x = a \cos(\theta - \alpha)$ and $y = b \cos(\theta - \beta)$,then $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\alpha - \beta)$ is equal to

If $3 \sin \alpha = 5 \sin \beta$,then $\tan \left(\frac{\alpha + \beta}{2}\right) \div \tan \left(\frac{\alpha - \beta}{2}\right) = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module