Question
$\text{If}\ \text{m}\sin\theta=\text{n}\sin(\theta+2\alpha),$ prove that $\tan(\theta+\alpha)\cot\alpha=\frac{\text{m+n}}{\text{m}-\text{n}}.$
✓
Answer
Given that $\text{m}\sin\theta=\text{n}\sin(\theta+2\alpha),$
We need to prove that $\tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\tan\alpha$
$\text{m}\sin\theta=\text{n}\sin(\theta+2\alpha)$
$\Rightarrow\ \frac{\sin(\theta+2\alpha)}{\sin\theta}=\frac{\text{m}}{\text{n}}$
Using Componendo - Dividendo, we have,
$\Rightarrow\ \frac{\sin(\theta+2\alpha)+\sin\theta}{\sin(\theta+2\theta)-\sin\theta}=\frac{\text{m+n}}{\text{m}-\text{n}}...(1)$
We know that,
$\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C+D}}{2}\cos\frac{\text{C}-\text{D}}{2}$
and
$\sin\text{C}-\sin\text{D}=2\cos\frac{\text{C+D}}{2}\sin\frac{\text{C}-\text{D}}{2}$
Applying the above formula in equation (1), we have,
$\frac{2\sin\frac{\theta+2\theta+\theta}{2}\cos\frac{\theta+2\theta-\theta}{2}}{2\cos\frac{\theta+2\theta+\theta}{2}\sin\frac{\theta+2\theta-\theta}{2}}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \frac{2\sin(\theta+\alpha)\cos\alpha}{2\cos(\theta+\alpha)\sin\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \frac{\tan(\theta+\alpha)}{\tan\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\times\tan\alpha$
Hence proved.
We need to prove that $\tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\tan\alpha$
$\text{m}\sin\theta=\text{n}\sin(\theta+2\alpha)$
$\Rightarrow\ \frac{\sin(\theta+2\alpha)}{\sin\theta}=\frac{\text{m}}{\text{n}}$
Using Componendo - Dividendo, we have,
$\Rightarrow\ \frac{\sin(\theta+2\alpha)+\sin\theta}{\sin(\theta+2\theta)-\sin\theta}=\frac{\text{m+n}}{\text{m}-\text{n}}...(1)$
We know that,
$\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C+D}}{2}\cos\frac{\text{C}-\text{D}}{2}$
and
$\sin\text{C}-\sin\text{D}=2\cos\frac{\text{C+D}}{2}\sin\frac{\text{C}-\text{D}}{2}$
Applying the above formula in equation (1), we have,
$\frac{2\sin\frac{\theta+2\theta+\theta}{2}\cos\frac{\theta+2\theta-\theta}{2}}{2\cos\frac{\theta+2\theta+\theta}{2}\sin\frac{\theta+2\theta-\theta}{2}}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \frac{2\sin(\theta+\alpha)\cos\alpha}{2\cos(\theta+\alpha)\sin\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \frac{\tan(\theta+\alpha)}{\tan\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$
$\Rightarrow\ \tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\times\tan\alpha$
Hence proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find three numbers in G.P. whose sum is $65$ and whose product is $3375$.
→Some schools send their students for extra training in Kabaddi, Cricket and Tennis to a sports standidium. There center charge fee is changed pen student for Coaching as well as 4 equipment and maintenances of the court. The information of students from each school is given below-
If $a, b, c$ are in A.P. and $a, b, d$ are in G.P., then prove that $a, a - b, d - c$ are in G.P.
→If for $\text{f}(\text{x})=\lambda\text{x}^2+\mu\text{x}+12,\text{f}'(\text{x})=15$ and $\text{f}'(\text{2})=11,$ then find $\lambda$and $\mu.$
→Find the general solutions of the following equations:
$\sin3\text{x}+\cos2\text{x}=0$
→$\sin3\text{x}+\cos2\text{x}=0$
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases:
conjugate axis is 5 and the distance between foci = 13
conjugate axis is 5 and the distance between foci = 13
A railway train is travelling on a circular curve of 1500 metres radius at the rate of 66km/ hr. Through what angle has it turned in 10 seconds?
There is a working women’s hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hail from the same town are graduates and 83% of the women who have come from the neighbouring town are also graduates. Find the probability that a woman selected at random is a graduate from the same town.
Find the length of the perpendicular from the point (4, -7) to the line joining the origin and the point of intersection of the lines 2x - 3y + 14 = 0 and 5x + 4y - 7 = 0.
Find the equation of the parabola, if
The focus is at (-6, - 6) and the vertex is at (-2, 2).
The focus is at (-6, - 6) and the vertex is at (-2, 2).