Evaluate the following: If A = 30^ , verify that: 2A= 1- ^2A 1+ ^… - Vidyadip

Question

Evaluate the following:
If $A = 30^\circ ,$ verify that:
$\cos2\text{A}=\frac{1-\tan^2\text{A}}{1+\tan^2\text{A}}$

✓

Answer

$\text{A}=30^\circ$
$\Rightarrow2\text{A}=2\times30^\circ=60^\circ$
$\cos2\text{A}=\cos60^\circ=\frac12$
$\frac{1-\tan^2\text{A}}{1+\tan^2\text{A}}=\frac{1-\tan^230^\circ}{1+\tan^230^\circ}$
$=\frac{1-\Big(\frac{1}{\sqrt{3}}\Big)^2}{1+\Big(\frac{1}{\sqrt{3}}\Big)^2}=\frac{\Big(1-\frac{1}{{3}}\Big)}{\Big(1+\frac13\Big)}$
$=\Big(\frac{2}{3}\Big)\times\frac34=\frac12$
$\therefore\ \cos2\text{A}=\frac{1-\tan^2\text{A}}{1+\tan^2\text{A}}$

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 "3 Marks Question" from Trigonometric Ratios of Some Particular Angles→Trigonometric Ratios of Some Particular Angles — all question types→Maths STD 10 question papers→Gujarat Board STD 10 question papers→Gujarat Board question paper generator→

Similar questions

$50$ circular plates each of diameter $14\ cm$ and thickness $0.5\ cm$ are placed one above the other to form a right circularcylinder. Find its total surface area.

Find the ratio in which the point $\text{P}\Big(\frac{3}{4},\frac{5}{12}\Big)$ divides the line segment joining the points $\text{A}\Big(\frac{1}{2},\frac{3}{2}\Big)$ and $B(2, -5).$

If the sum of the first n terms of an $A.P.$ is $4 n-n^2$, what is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the nth term.

Prove that $\frac{1}{{\sqrt 2 }}$ is irrational.

Find the ratio of the area of the circle circumscribing a square to the area of the circle inscribed in the square.

In $\triangle\text{ABC and }\triangle\text{DEF},$ it is being given that: $AB = 5\ cm, BC = 4\ cm$ and $CA = 4.2\ cm; DE = 10\ cm, EF = 8\ cm$ and $FD = 8.4\ cm.$ If $\text{AL}\perp\text{BC}$ and $\text{DM} \perp \text{EF,}$ find $AL : DM.$

To find out the concentration of $SO_2$ in the air $($in parts per million, i.e. $ppm),$ the data was collected for $30$ localities in a certain city and is presented below:

Concentration of $SO_2$ (in ppm)	Frequency
$0.00-0.04$	$4$
$0.04-0.08$	$9$
$0.08-0.12$	$9$
$0.12-0.16$	$2$
$0.16-0.20$	$4$
$0.20-0.24$	$2$

Find the mean concentration of $SO_2$ in the air.

State Pythagoras theoram.

The side of a square is $10\ cm.$ Find the area of circumscribed and inscribed circles.

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.
$\sin \theta =\frac{\sqrt{3}}{2}$