If ( + )=m ( - ), then prove that = 1-m 1+m . [Hint: Express ( + ) ( - ) = m 1 and apply Componendo and Dividendo]

Given that: ( + )=m ( - ) ( + ) ( - ) = m 1 Using componendo and dividend rule, we get ( + )+ ( - ) ( + )- ( - ) = m+1 m-1 [ - =-2 ( C+D 2 ) ( C-D 2 ) ] [ + =2 ( C+D 2 ) ( C-D 2 ) ] 2 ( + + - 2 ). ( + - + 2 ) -2 ( + + - 2 ). ( + - + 2 ) = m+1 m-1 - = m+1 m-1 - . = m+1 m-1 - = m+1 m-1 =- 1+m 1-m (1+m)=(1-m) = 1-m 1+m . Hence proved.

If ( + )=m ( - ), then prove that = 1-m 1+m . [Hint: Express ( + …

Download App

Question

If $\cos(\theta+\phi)=\text{m}\cos(\theta-\phi),$ then prove that $\tan \theta=\frac{1-\text{m}}{1+\text{m}}\cot\phi.$
[Hint: Express $\frac{\cos(\theta+\phi)}{\cos(\theta-\phi)}=\frac{\text{m}}{1}$ and apply Componendo and Dividendo]

✓

Answer

Given that: $\cos(\theta+\phi)=\text{m}\cos(\theta-\phi)$
$\Rightarrow\frac{\cos(\theta+\phi)}{\cos(\theta-\phi)}=\frac{\text{m}}{1}$
Using componendo and dividend rule, we get
$\frac{\cos(\theta+\phi)+\cos(\theta-\phi)}{\cos(\theta+\phi)-\cos(\theta-\phi)}=\frac{\text{m}+1}{\text{m}-1}$
$\Big[\cos\text{C}-\cos\text{D}=-2\sin\Big(\frac{\text{C+D}}{2}\Big)\sin\Big(\frac{\text{C}-\text{D}}{2}\Big)\Big]$
$\Big[\cos\text{C}+\cos\text{D}=2\cos\Big(\frac{\text{C+D}}{2}\Big)\cos\Big(\frac{\text{C}-\text{D}}{2}\Big)\Big]$
$\Rightarrow\frac{2\cos\Big(\frac{\theta+\phi+\theta-\phi}{2}\Big).\cos\Big(\frac{\theta+\phi-\theta+\phi}{2}\Big)}{-2\sin\Big(\frac{\theta+\phi+\theta-\phi}{2}\Big).\sin\Big(\frac{\theta+\phi-\theta+\phi}{2}\Big)}=\frac{\text{m}+1}{\text{m}-1}$
$\Rightarrow\frac{\cos\theta\cos\phi}{-\sin\theta\sin\phi}=\frac{\text{m}+1}{\text{m}-1}\Rightarrow-\cot\theta.\cot\phi=\frac{\text{m}+1}{\text{m}-1}$
$\Rightarrow\frac{-\cot\phi}{\tan\theta}=\frac{\text{m}+1}{\text{m}-1}=-\frac{1+\text{m}}{1-\text{m}}$
$\Rightarrow\tan\theta(1+\text{m})=(1-\text{m})\cot\phi$
$\Rightarrow\tan\theta=\frac{1-\text{m}}{1+\text{m}}\cot\phi.$ 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.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Trigonometric Functions — Maths STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions