Download App

Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
Prove the following identities:
$(1+\tan\theta+\cot\theta)(\sin\theta-\cos\theta)=\Big(\frac{\sec\theta}{\text{cosec}^2\theta}-\frac{\text{cosec}\theta}{\sec^2\theta}\Big)$
✓

Answer

$\text{LHS}=(1+\tan\theta+\cot\theta)(\sin\theta-\cos\theta)$
$=\Big(1+\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}\Big)(\sin\theta-\cos\theta)$
$=\Big(\frac{\cos\theta\sin\theta+\sin^2\theta+\cos^2\theta}{\cos\theta\sin\theta}\Big)(\sin\theta-\cos\theta)$
$=\frac{(\cos\theta\sin\theta+1)}{\cos\theta\sin\theta}(\sin\theta-\cos\theta)$
$\text{RHS}=\Big(\frac{\sec\theta}{\text{cosec}^2\theta}-\frac{\text{cosec }\theta}{\sec^2\theta}\Big)$
$=\Bigg(\frac{\frac{1}{\cos\theta}-\frac{1}{\sin\theta}}{\frac{1}{\sin^2\theta}\frac{1}{\cos^2\theta}}\Bigg)$
$=\Big(\frac{\sin^2\theta}{\cos\theta}-\frac{\cos^2\theta}{\sin\theta}\Big)=\frac{\sin^3\theta-\cos^3\theta}{\cos\theta\sin\theta}$
$=\frac{(\sin\theta-\cos\theta)\big(\sin^2\theta+\cos^2\theta+\cos\theta\sin\theta\big)}{\cos\theta\sin\theta}$ $\Big[\because\text{a}^3-\text{b}^3=(\text{a}-\text{b})\big(\text{a}^2+\text{ab}+\text{b}^2\big)\Big]$
$=\frac{(\sin\theta-\cos\theta)(1+\cos\theta\sin\theta)}{\cos\theta\sin\theta}$
$\therefore\ \text{R.H.S.}=\text{L.H.S.}$

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 the given figure, three sectors of a circle of radius 7cm, making angles of 60°, 80° and 40° at the centre are shaded. Find the area of the shaded region.
→
A street light bulb is fixed on a pole 6m above the level of the street. If a woman of height 1.5m casts a shadow of 3m, find how far she is away from the base of the pole.
→
From the following frequency distribution, prepare the 'More Then Ogive'.
Score
Number of candidates
400-450
20
450-500
35
500-550
40
550-600
32
600-650
24
650-700
27
700-750
18
750-800
24
Total
230
Also find the median.
→
Find the leash number of square tiles required to pave the ceiling of a room 15m 17cm long and 9m 2cm broad.
→
If $(m+1)^{\text {th }}$ term of an A.P is twice the $(n+1)^{\text {th }}$ term, prove that $(3 m+1)^{\text {th }}$ term is twice the $(m+n+1)^{\text {th }}$ term.
→
An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are $45^{\circ}$ and $60^{\circ}$ respectively. Find the width of the river. [Use $\sqrt{3}=1.732$ ]
→
Solve the following system of equations by the method of cross-multiplication:
5ax + 6by = 28,
3ax + 4by = 18.
→
A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of getting:
  1. A queen.
  2. A diamond.
  3. A king or an ace.
  4. A red ace.
→
Amita buys some books for ₹1920. If she had bought 4 more books for the same amount each book would cost her ₹ 24 less. How many books did she buy? What was the initial price of one book?
→
The perimeter of sector OACB of the circle centred at O and of radius 24, is 73.12 cm.
Image
(i) Find the central angle $\angle A O B$.
(ii) Find the area of the minor segment ACB. (Use $\pi=3.14$ and $\sqrt{3}=1.73$ )
→