Download App

Trigonometric Identities — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsTrigonometric Identities4 Marks
Question
Prove the following trigonometric identities.
$(\sec\text{A}-\text{cosec A})(1+\tan\text{A}+\cot\text{A})=\tan\text{A}\sec\text{A}-\cot\text{A}\text{ cosec A}$
✓

Answer

$\text{L.H.S}=(\sec\text{A}-\text{cosec A})(1+\tan\text{A}+\cot\text{A})$
$=\Big[\frac{1}{\cos\text{A}}-\frac{1}{\sin\text{A}}\Big]\Big[1+\frac{\sin\text{A}}{\cos\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}\Big]$
$=\Big[\frac{\sin\text{A}-\cos\text{A}}{\sin\text{A}\cos\text{A}}\Big]\Big[\frac{\cos\text{A}\sin\text{A}+\sin^2\text{A}+\cos^2\text{A}}{\sin\text{A}\cos\text{A}}\Big]$
$=\frac{(\sin\text{A}-\cos\text{A})(\sin^2\text{A}+\cos\text{A}\sin\text{A}+\cos^2\text{A})}{\sin^2\text{A}\cos^2\text{A}}$
$=\frac{(\sin^3\text{A}-\cos^3\text{A})}{\sin^2\text{A}\cos^2\text{A}} \big[\therefore\ (\text{a}-\text{b})(\text{a}^2+\text{ab})+\text{b}=(\text{a}^3-\text{b}^2)\big]$
$=\text{R.H.S}=\tan\text{A}\sec\text{A}-\cot\text{A}\text{ cosec A}$
$=\frac{\sin\text{A}}{\cos\text{A}}\times\frac{1}{\cos\text{A}}-\frac{\cos\text{A}}{\sin\text{A}}\times\frac{1}{\sin\text{A}}$
$=\frac{\sin\text{A}}{\cos^2\text{A}}-\frac{\cos\text{A}}{\sin^2\text{A}}$
$=\frac{\sin^3\text{A}-\cos^3\text{A}}{\sin^2\text{A}\cos^2\text{A}}$
$\text{L.H.S}=\text{R.H.S}$
Hence proved.

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

If the roots of the equation $(b-c) x^2+(c-a) x+(a-b)=0$ are equal, then prove that $2 b=a+c$.
→
Prove the following identities:
$\frac{\sin\theta}{(\cot\theta+\text{cosec }\theta)}-\frac{\sin\theta}{(\cot\theta-\text{cosec }\theta)}=2$
→
$ABCD$ is a square. $F$ is the mid-point of $AB. BE$ is one third of $BC.$ If the area of $\triangle\text{FBE}=108\text{cm}^2,$ find the length of $AC.$
→
Find the area of a triangle whose vertices are,
$(\text{at}_1^2, 2\text{at}_1), (\text{at}_2^2, 2\text{at}_2)$ and $(\text{at}_3^2, 2\text{at}_3).$
→
In the following data the median of the runs scored by 60 top batsmen of the world in one-day international cricket matches is 5000. Find the missing frequencies x and y.
Runs scored
2500-3500
3500-4500
4500-5500
5500-6500
6500-7500
7500-8500
Number of batsmen
5
x
y
12
6
2
→
How many lead balls, each of radius 1cm, can be made from a sphere of radius 8cm?
→
The sum of first three terms of an AP is 48. If the product of first and second terms exceeds 4 times the third term by 12. Find the AP.
Hint: Let these terms be (a - b), a, (a + d).
→
The sum of the first $7$ terms of an AP is 182. If its $4^{th}$​​​​​​​ and $17^{th}​​​​​​​$​​​​​​​ terms are in the ratio $1 : 5$, find the AP.
→
In $\Delta$ABC, seg DE || side BC. If 2A ($\Delta$ADE) = A($\square$DBCE). Show that BC = $\sqrt{3} \times$ DE
→
In the figure quadrilateral ABCD is cyclic, If m(arc BC) = 90° and ∠DBC = 55°. Then find the measure of ∠BCD .Image
→