If 2 2 = 2 , prove that =3+5 5+3 - Vidyadip

Question

If $2\tan\frac{\alpha}{2}=\tan\frac{\beta}{2},$ prove that $\cos\alpha=\frac{3+5\cos\beta}{5+3\cos\beta}$

✓

Answer

We have,
$2\tan\frac{\alpha}{2}=\tan\frac{\beta}{2}$
$\Rightarrow\frac{\tan\frac{\alpha}{2}}{\tan\frac{\beta}{2}}=\frac{1}{2}$
Let $\tan\frac {\alpha}{2}=\text{k}$ and $\tan\frac{\beta}{2}=2\text{k}$
Then,
$\cos\alpha=\frac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}=\frac{1-\text{k}^2}{1+\text{k}^2}\ .....(\text{A})$
Also,
$\frac{3+5\cos\beta}{5+3\cos\beta}=\frac{3+5\Bigg(\frac{1-\tan^2\frac{\beta}{2}}{1+\tan^2\frac{\beta}{2}}\Bigg)}{5+3\Bigg(\frac{1-\tan^2\frac{\beta}{2}}{1+\tan^2\frac{\beta}{2}}\Bigg)}$
$=\frac{3+5\Big(\frac{1-4\text{k}^2}{1+4\text{k}^2}\Big)}{5+3\Big(\frac{1-4\text{k}^2}{1+4\text{k}^2}\Big)}$
$=\frac{8-8\text{k}^2}{8+8\text{k}^2}=\frac{1-\text{k}^2}{1+\text{k}^2}\ .....(\text{B})$
From(A) & (B)
$\cos\alpha=\frac{3+5\cos\beta}{5+3\cos\beta}$

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 Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

A student obtained the mean and standard deviation of 100 observations as 40 and 5.1 respectively. It was later found that one observation was wrongly copied as 50, the correct figure being 40. Find the correct mean and S.D.

If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other,and $\phi$ is their difference,the show that $\sin\theta=\frac{\lambda+1}{\lambda-1}\sin\phi$

$2\text{x}^4+5\text{x}^3+7\text{x}^2-\text{x}+41,$ when $\text{x}=-2-\sqrt{3}\text{i}$

Find the equation of the hyperbola whose
Focus is at (4, 2) centre at (6, 2) and e = 2.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos\text{ax}-\cos\text{bx}}{\cos\text{cx}-1}$

In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:

The vowels always occupy even places?
The relative order of vowels and consonants do not alter?

A farmer buys a used tractor for ₹ 12000. He pays ₹ 6000 cash and agrees to pay the balance in annual instalments of ₹ 500 plus 12% interest on the unpaid amount. How much the tractor cost him?

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{6}}\frac{\text{cosec}^2\text{ x}-3}{\text{cosec}\text{ x}-2}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{-1}}\frac{\text{x}^{3}+1}{\text{x}+1}$