Prove the following identities: ^2 (1+ ^2 ) + ^2 (1+ ^2 ) =1 - Vidyadip

Question

Prove the following identities:
$\frac{\tan^2\theta}{\big(1+\tan^2\theta\big)}+\frac{\cot^2\theta}{\big(1+\cot^2\theta\big)}=1$

✓

Answer

$\text{L.H.S.}=\frac{\tan^2\theta}{\big(1+\tan^2\theta\big)}+\frac{\cot^2\theta}{\big(1+\cot^2\theta\big)}$
$=\frac{\tan^2\theta}{\sec^2\theta}+\frac{\cot^2\theta}{\text{cosec}^2\theta}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta$ and $\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\Big]$
$=\frac{\frac{\sin^2\theta}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}+\frac{\frac{\cos^2\theta}{\sin^2\theta}}{\frac{1}{\sin^2\theta}}$
$=\sin^2\theta+\cos^2\theta=1$
$=\text{R.H.S.}$
Hence, LHS = RHS.

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 "4 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following equations by using the method of completing the square:
$\sqrt3\text{x}^2+10\text{x}+7\sqrt3=0$

Show that A (-4, -7),B (-1, 2), C (8, 5) and D (5, -4) are the vertices of a parallelogram.

From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP = diameter of the circle, show that $\triangle\text{APB}$ is equilateral.

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for $4$ hours and the pipe of smaller diameter for $9$ hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes $10$ hours more than the pipe of larger diameter to fill the pool?

The radii of the ends of a bucket $30\ cm$ high are $21\ cm$ and $7\ cm$. Find its capacity in litres and the amount of sheet required to make this bucket.

Solve the following quadratic equations by factorization:
$\frac{3}{\text{x}+1}-\frac{1}{2}=\frac{2}{3\text{x}-1},$ $\text{x}\neq-1,\frac{1}{3}$

Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is $800m^2$?
If so, find its length and breadth.

Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:

2x - 3y -7 = 0.

$x+2 y+4=0 ; 3 x=-4 y-16$

Kargil’s temperature was recorded in a week from Monday to Saturday. All readings were in A.P. The sum of temperatures of Monday and Saturday was 5° C more than sum of temperatures of Tuesday and Saturday. If temperature of Wednesday was – 30° celsius then find the temperature on the other five days.