Download App

Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
Prove the following trigonometric identities.
If $\text{a}\cos\theta+\text{b}\sin\theta=\text{m and a}\sin\theta-\text{b}\cos\theta=\text{n},$ prove that $a^2 + b^2 = m^2 + n^2.$
✓

Answer

Given,
$\text{a}\cos\theta+\text{b}\sin\theta=\text{m},$
$\text{a}\sin\theta-\text{b}\cos\theta=\text{n}$
We have to prove $a^2 + b^2 = m^2 + n^2$​​​​​​​
We know that, $\sin^2\theta+\cos^2\theta=1$
Now, squaring and adding the two equations, we get
$(\text{a}\cos\theta+\text{b}\sin\theta)^2+(\text{a}\sin\theta-\text{b}\cos\theta)^2=\text{m}^2+\text{n}^2$
$\Rightarrow \big(\text{a}^2\cos^2\theta+2\text{ab}\sin\theta\cos\theta+\text{b}^2\sin^2\theta\big)$
$+\big(\text{a}^2\sin^2\theta-2\text{ab}\sin\theta\cos\theta+\text{b}^2\cos^2\theta\big)=2$
$\Rightarrow\ \text{a}^2(\cos^2\theta+\sin^2\theta)+\text{b}^2(\sin^2\theta+\cos^2\theta)=\text{m}^2+\text{n}^2$
$\Rightarrow\ \text{a}^2+\text{b}^2=\text{m}^2+\text{n}^2$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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 Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students. Which value is shown in this Question?
→
Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the coordinates of the point of division.
→
Find the sum of,
All 3 - digit natural numbers which are divisible by 13.
→
If the sum of first p term of an AP is $(ap^2+ bp)$, find its common difference.
→
If sum of the $3^{\text {rd }}$ and the $8^{\text {th }}$ terms of an AP is 7 and the sum of the $7^{\text {th }}$ and the $14^{\text {th }}$ terms is -3 , find the $10^{\text {th }}$ term.
→
Find a relation between x and y, if the points A(2, 1), B(x, y), and C(7, 5) are collinear.
→
Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not:
2x - 3y = 6, x + y = 1.
→
In the given figure, PQ = 24cm, PR = 7cm and O is the centre of the circle. Find th area of the shaded region. $\big[\text{Take }\pi=3.14\big]$
→
In $\triangle\text{ABC},\ \angle\text{ABC}=135^\circ.$ Prove that $AC^2 = AB^2 + BC^2 + 4$ $\text{ar}(\triangle\text{ABC}).$
→
The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of houses preceding the house numbered X is equal to sum of the numbers of houses following X.
→