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1: Real Numbers
Algebra
2: Polynomials
3: Linear Equations in Two Variables
4: Quadratic Equations
5: Arithmetic Progression
Coordinate Geometry
6: Coordinate Geometry
Geometry
7: Triangles
8: Circles
9: Constructions
Trigonometry
10: Trignometric Ratios
11: T-Ratios of Some Particular Angles
12: Trigonometric Ratios of Some Complemantary Angles
▶ 13: Trigonometric identities
14: Heights and Distances
Mensuration
15: Perimeter And Area of Plane Figures
16: Area of Circle, Sector and Segment
17: Volumes and Surface Areas of Solids
Statistics and Probability
18: Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive
19: Probability
Chapter 20: Additional Questions
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Solutions for Chapter 13: Trigonometric identities
Below listed, you can find solutions for Chapter 13 of CBSE, Karnataka Board R.S. Aggarwal for माठेमटिक्स [इंग्रजी] इयत्ता १०.
R.S. Aggarwal solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० 13 Trigonometric identities EXERCISE 13A [Pages 617 - 618]
Prove the following identities:
sin4θ – cos4θ = 1 – 2cos2θ
Prove the following identities:
`(sin^4θ + cos^4θ)/(1 - 2sin^2θ cos^2θ) = 1`
Prove the following identities:
`1 + (cot^2θ)/(1 + "cosec" θ) = "cosec" θ`
Prove the following identities:
(sec A – cos A) (cot A + tan A) = sec A tan A
Prove the following identities:
`(1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2`
Prove the following identities:
sin A (1 + tan A) + cos A (1 + cot A) = sec A + coseс А
Prove the following identities:
sec4θ – tan4θ = 1 + 2 tan2θ
Prove the following identities:
`sin theta/((cot theta + "cosec" theta)) - sin theta/((cot theta - "cosec" theta)) = 2`
Prove the following identities:
`("cosec" A - sin A)/("cosec" A + sin A) = (sec^2A - tan^2A)/(sec^2A + tan^2A)`
Prove the following identities:
sin2θ tan θ + cos2θ cot θ + 2 sin θ cos θ = tan θ + cot θ
Prove the following identities:
`(tan θ + sec θ - 1)(tan θ + sec θ + 1) = (2 sin θ)/(1 - sin θ)`
Prove the following identities:
(1 + cot A + tan A) (sin A – cos A) = sin A tan A – cot A cos A
Prove the following identities:
`(sec θ + tan θ)/(sec θ - tan θ) = ((1 + sin θ)/cos θ)^2`
Prove the following identities:
`sec^2θ - (sin^2θ - 2sin^4θ)/(2cos^4θ - cos^2θ) = 1`
Prove that `sin A/(sec A + tan A - 1) + cos A/("cosec" A + cot A - 1) = 1`.
Prove the following identities:
`(tan A + sin A)/(tan A - sin A) = (sec A + 1)/(sec A - 1)`
Prove the following identities:
`(cot A - cos A)/(cot A + cos A) = ("cosec" A - 1)/("cosec" A + 1)`
Prove the following identities:
`(1 - sin θ)/(1 + sin θ) = (sec θ - tan θ)^2`
Prove the following identities:
`(1 + cos theta)/(1 - cos theta) = ("cosec" theta + cot theta)^2`
Prove the following identities:
`((sec θ - tan θ))/((sec θ + tan θ)) = (1 + 2 tan^2θ - 2 sec θ tan θ)`
Prove the following identities:
`1 + (tan^2θ)/((1 + sec θ)) = sec θ`
Prove the following identities:
`(cos^3θ + sin^3θ)/(cos θ + sin θ) + (cos^3θ - sin^3θ)/(cos θ - sin θ) = 2`
Prove the following identities:
`(1 + cos theta - sin^2 theta )/(sin theta (1 + cos theta)) = cot theta`
Prove the following identities:
`("cosec" theta + cot theta)/("cosec" theta - cot theta) = ("cosec" theta + cot theta )^2 = 1 + 2 cot^2 theta + 2 "cosec" theta cot theta`
Prove the following identities:
`(sec theta + tan theta)/(sec theta - tan theta) = (sec theta + tan theta)^2 = 1 + 2 tan^2 theta + 2 sec theta tan theta`
Prove the following identities:
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta )/(cos theta)`
Prove the following identities:
`(sin theta + 1 - cos theta)/(cos theta - 1 + sin theta) = (1 + sin theta)/(cos theta)`
Prove the following identities:
`(cos theta "cosec" theta - sin theta sec theta )/(cos theta + sin theta) = "cosec" theta - sec theta`
Prove the following identities:
`(1 + tan theta + cot theta)(sin theta - cos theta) = ((sec theta)/("cosec"^2 theta) - ("cosec" theta)/(sec^2 theta))`
Prove the following identities:
`(cot^2 theta (sec theta - 1))/((1 + sin theta)) + (sec^2 theta(sin theta - 1))/((1 + sec theta)) = 0`
Show that none of the following is an identity:
`cos^2theta + cos theta = 1`
Show that none of the following is an identity:
`sin^2 theta + sin theta = 2`
Show that none of the following is an identity:
`tan^2 theta + sin theta = cos^2 theta`
R.S. Aggarwal solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० 13 Trigonometric identities EXERCISE 13В [Pages 628 - 629]
If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that (m2 + n2) = (a2 + b2).
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x2 – y2) = (a2 – b2).
If `(x/a sin theta - y/b cos theta) = 1` and `(x/a cos theta + y/b sin theta) = 1`, prove that `(x^2/a^2 + y^2/b^2) = 2`.
If x = a cos3θ and y = b sin3θ, prove that `(x/a)^(2/3) + (y/b)^(2/3) = 1`.
If x = b sec3θ and y = a tan3θ, prove that `(x/b)^(2/3) - (y/a)^(2/3) = 1`.
If (tan θ + sin θ) = m and (tan θ – sin θ) = n, prove that (m2 – n2)2 = 16 mn.
If (cot θ + tan θ) = m and (sec θ – cos θ) = n, prove that `(m^2 n)^(2//3) - (mn^2)^(2//3) = 1`.
If (cosec θ – sin θ) = a3 and (sec θ – cos θ) = b3, prove that a2b2(a2 + b2) = 1.
If x = (sec A + sin A) and y = (sec A – sin A), prove that `(2/(x + y))^2 + ((x - y)/2)^2 = 1`.
If `m = (cos θ - sin θ)` and `n = (cos θ + sin θ)`, show that `sqrt(m/n) + sqrt(n/m) = 2/sqrt(1 - tan^2θ)`.
If `(cos θ - sin θ) = sqrt(2) sin θ` then prove that `(cos θ + sin θ) = sqrt(2) cos θ`.
If `(sec θ - tan θ) = sqrt(2) tan θ` then prove that `(sec θ + tan θ) = sqrt(2) sec θ`.
If (sec θ + tan θ) = p then show that `(sec θ - tan θ) = 1/p`. Hence, show that `cos θ = (2p)/(p^2 + 1)` and `sin θ = (p^2 - 1)/(p^2 + 1)`.
If (cosec A + cot A) = m then prove that `(m^2 - 1)/(m^2 + 1) = cos A`.
If (sec A – tan A) = x then prove that `(1 + x^2)/(1 - x^2)` = cosec A.
If `(sin θ + cos θ) = sqrt(2)`, prove that (tan θ + cot θ) = 2.
R.S. Aggarwal solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० 13 Trigonometric identities MULTIPLE-CHOICE QUESTIONS (MCQ) [Pages 630 - 632]
Choose the correct answer in each of the following questions:
If sin A + sin2A = 1 then (cos2A + cos4A) = ?
`1/2`
1
2
3
`(1 + tan^2A)/(1 + cot^2A)` = ?
–1
sec2A
tan2A
cot2A
(sec A + tan A)(1 – sin A) = ?
sin A
cos A
sec A
cosec А
(1 + tan θ + sec θ) (1 + cot θ – cosec θ) = ?
–1
0
1
2
If sin θ – cos θ = 0 then the value of (sin4θ + cos4θ) is ______.
`1/4`
`1/2`
`3/4`
1
If cos 9α = sin α and 9α < 90°, then the value of tan 5α is ______.
`1/sqrt(3)`
`sqrt(3)`
1
0
The value of `(tan^2θ - sec^2θ)/(cot^2θ - "cosec"^2θ)` is ______.
–1
`1/2`
`(-1)/2`
1
`sqrt((1 - sin A)/(1 + sin A)` = ?
sec A – tan A
sec A + tan A
sec A tan A
None of these
`sqrt((1 + cos A)/(1 - cos A)` = ?
cosec A – cot A
cosec A + cot A
cosec A cot A
sin A tan A
(cosec θ – cot θ)2 = ?
`(1 + cos θ)/(1 - cos θ)`
`(1 - cos θ)/(1 + cos θ)`
`(1 + sin θ)/(1 - sin θ)`
`(1 - sin θ)/(1 + sin θ)`
If (tan θ + cot θ) = 5 then the value of (tan2θ + cot2θ) is ______.
23
24
25
27
If `tan θ = a/b` then `((cos θ + sin θ))/((cos θ - sin θ))` = ?
`(a + b)/(a - b)`
`(a - b)/(a + b)`
`(b + a)/(b - a)`
`(b - a)/(b + a)`
If `(cos θ + sec θ) = 5/2` then the value of (cos2θ + sec2θ) is ______.
`21/4`
`17/4`
`29/4`
`33/4`
If `tan θ = 8/15` then cosec θ = ?
`15/17`
`17/15`
`17/8`
`8/17`
If 5 tan θ = 3 then `((5 sin θ - cos θ)/(5 sin θ + cos θ))` = ?
`2/3`
`1/3`
`1/2`
`3/5`
In a ΔAВС, ∠C = 90°. Then, the value of cos (A + B) is ______.
0
1
`1/2`
`3/5`
If cos A + cos2 A = 1 then sin2 A + sin4 A = ?
1
2
3
4
If 3 cot θ = 4 then the value of `((5 sin θ + 3 cos θ)/(5 sin θ - 3 cos θ))` = ?
`1/3`
3
`1/9`
9
If 2x = sec A and `2/x = tan A` then `2(x^2 - 1/x^2)` = ?
`1/2`
`1/4`
`1/8`
`1/16`
If 3x = cosec x and `3/x = cot x` then `3(x^2 - 1/x^2)` = ?
`1/27`
`1/81`
`1/3`
`1/9`
If `tan θ = sqrt(3)` then sec θ = ?
2
`1/2`
`sqrt(3)/2`
`2/sqrt(3)`
Solutions for 13: Trigonometric identities
R.S. Aggarwal solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० chapter 13 - Trigonometric identities
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