If 2 =3 , prove that ( - )= 2 5- 2 - Vidyadip

Question

If $2\tan\alpha=3\tan\beta,$ prove that $\tan(\alpha-\beta)=\frac{\sin2\beta}{5-\cos2\beta}$

✓

Answer

We have,
$2\tan\alpha=3\tan\beta$
$\Rightarrow\frac{\tan\alpha}{\tan\beta}=\frac{3}{2}$
Let $\tan\alpha=3\text{k}$ and $\tan\beta=2\text{k}$
Also,
$\frac{\sin2\beta}{5-\cos2\beta}=\frac{\frac{2\tan\beta}{1+\tan^2\beta}}{5-\Big(\frac{1-\tan^\beta}{1+\tan^2\beta}\Big)}$
$=\frac{\frac{2.2\text{k}}{1+4\text{k}^2}}{5-\Big(\frac{1-4\text{k}^2}{1+4\text{k}^2}\Big)}$
$=\frac{4\text{k}}{5+20\text{k}^2-1+4\text{k}^2}$
$=\frac{4\text{k}}{4+24\text{K}^2}=\frac{\text{K}}{1+6\text{k}^2}\ ...\text{(B)}$
From (A) & (B)
$\tan(\alpha-\beta)=\frac{\sin2\beta}{5-\cos2\beta}$

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 Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove the following by using the principle of mathematical induction for all n ∈ N:$\text{a}+\text{ar}+\text{ar}^2+...+\text{ar}^{\text{n-1}}=\frac{\text{a}(\text{r}^\text{n}-1)}{\text{r}-1}.$

Find the equation of lines passing through $(1, 2)$ and making angle $30^\circ$ with $y-$axis.

There are $60$ students in a class. The following is the frequency distribution of the marks obtained by the students in a test:

Marks	$0$	$1$	$2$	$3$	$4$	$5$
Frequency	$x - 2$	$x$	$x^2$	$(x + 1)^2$	$2x$	$x + 1$

where $x$ is a positive integer. Determine the mean and standard deviation of the marks.

$\frac{\cos^2\text{B}-\cos^2\text{C}}{\text{b + c}}+\frac{\cos^2\text{C}-\cos^2\text{A}}{\text{c + a}}+\frac{\cos^2\text{A}-\cos^2\text{B}}{\text{a + b}}=0$

Find the equations to the straight lines passing through the point (2, 3) and inclined at and angle of 45° to the line 3x + y - 5 = 0.

The sides of a square are $x = 6, x = 9, y = 3$ and $y = 6.$ Find the equation of a circle drawn on the diagonal of the square as its diameter.

Let A and B be two stes such that: $\text{n(P)} = 20, \text{n(A}\cup\text{B)=42 and n(A}\cap\text{B})=4.$ Find:
$\text{n(A} - \text{B)}.$

Solve the following systems of inequations graphically:
$\text{x}+2\text{y}\leq40,3\text{x}+\text{y}\geq30,4\text{x}+3\text{y}\geq60,\text{x}\geq0,\text{y}\geq0$

Use the Principle of Mathematical Induction in the following Exercis.
Prove that $\frac{1}{\text{n}+1}+\frac{1}{\text{n}+2}+\ .....\ +\frac{1}{2\text{n}}>\frac{13}{24},$ for all natural numbers n > 1.

Solve the following system of inequalities.
$\frac{2\text{x}+1}{7\text{x}-1}>5, \frac{\text{x}+7}{\text{x}-8}>2$