MULTIPLE AND SUBMULTIPLE
Short Questions:
i) If sinx + cosx=p, find sin2x. p²-1
ii) If sinx + cosx=. √2, find cos2x. 0
iii) If tanx= 1/3, find sin2x, cos2x, tan2x. 3/5,4/5,3/4
iv) Show cos2x + tanx sin2x =1.
v) If cos2x = 4/5, find tanx and cotx. ±1/3,±3
vi) Evaluate: (cos³x - cos3x)secx + (sin³x + sin3x)cosecx. 3
vii) If 0≤ x ≤ π/4 and sin2x= 4/5, find tanx. 1/2
viii) If 5sin²x + 3cos²x=4, find cos2x, sin2x. 0,1
ix) Show: 2 sin(π/8)= √{2 - √2}.
x) If sinx = - 4/5, find cos(x/2), sin(x/2), where x is an angle of third quadrant. -1/√5,2/√5
xi) If cosx=4/5, find cos2x, sin2x. 7/25, ±24/25
xii) If cotx - tanx = 2, find cot2x. 1
xiii) Show, 4(cos³10+ sin³20)= 3(cos10+ sin20).
xiv) If tanx tan 3x=1, find tan2x. ±1
xv) Show, 1/(sin10) - √3/(cos 10)= 4.
xvi) Find the maximum value of cos³x sinx - sin³x cosx. 1/4
xvii) If tan²β = (1- sinα)/(1+ sinα), Show α + 2β =π/2.
xviii) If cosx = 3/4, show 32sin(x/2) sin(5x/2)= 11.
xix) If cosx + sinx =√2 cosx, show, sin4x =1.
xx) Simplify: cos²(π/2 - x) - sin(2π/3 - x) sin(x - π/3). 3/4
xxi) If cos⁶x + sin⁶x + k sin²2x=1, find the value of k. 3/4
xxii) Show, (1+ cos(π/8))(1+ cos(3π/8))(1+ cos(5π/8))(1+ cos(7π/8)) = 1/8.
xxiii) Show, (2sinx)/(sin3x) + (tanx)/(tan3x)= 1.
Choose the Correct Option:
i) The value of 6 sin20- 8 sin³20 is
a) -1 b) 1 c) -√3 d) √3
ii) If tanx= b/a, then a cos2x + b sin2x equal to
a) a b) b c) a+ b d) none
iii) If tan(x/2)= 7, the value of 4sinx - 3 cosx is
a) 5 b) 4 c) 3 d) none
iv) If sinx - cosx =√2 sinx, the value of cot2x is
a) -1 b) 0 c) 1 d) √2
v) If cosx = -4/5 and sin2x = 24/25, the quadrant in which x lies, is
a) 1st quadrant b) 2nd quadrant c) 3rd quadrant d) 4th quadrant
vi) If π/2 < x < π and 3 sin²x + 5 cos²x = 4, the value of sin2x is
a) 1 b) 0 c) -1 d) none
vii) The value of sin (π/10) sin (13π/10) is
a) -1/4 b) 1/4 c) -√5/4 d) √5/4
viii) If 2 sec2x = tany + cot y, then one of the values of x + y is
a) π b) π/2 c) π/4 d) none
ix) If tan(x/2)= 2, then tanx will be
a) -5/3 b) -4/3 c) -3/5 d) 4/5
x) If 180°< x < 270° and sinx = - 3/5, then tan(x/2) is
a) -3 b) -1/3 c) -3 or -1/3 d) none
i d ii a iii b iv c v c vi c vii a viii c ix b x a
General Questions
1) Show, secx + tanx = tan(π/4 + x/2).
2) If tan²x = 1+ 2 tan²y, then show that
a) cos2y = 1+ 2 cos2x.
b) cos2x + sin²y = 0.
3) Show, tanx + 2 tan2x + 4 tan4x + 8 cot8x = cotx.
4) Show, (3+ cos4x)/(1- cos4x)= (1/2) (cot²x + tan²x).
5) Evaluate:
a) cos(π/5) cos(3π/5). -1/4
b) cos(π/5) - cos(2π/5). 1/2
6) Show, cos²(A - 60)+ cos²A+ cos²(A+ 60)= 3/2.
7) If x, y are acute angles and cos2x = (3 cos2y -1)/(3- cos2y), then show that, tanx = √2 tany.
8) Show, cos²40° cos²80° + cos²80° cos²20+ cos²20° cos²40° = 9/16.
9) If tanx= (tany + tanz)/(1+ tany tanz) show that sin2x= (sin2y + sin2z)/(1+ sin2y sin2z).
10) If tanx = 1/7 and tan y= 1/3, then show that x + 2y =π/4.
11) Show: tan 9 - tan 27 - tan 63 + tan 81=4.
12) Show cos⁴(π/16) + cos⁴(3π/16) + cos⁴(5π/16) + cos⁴(7π/16) =3/2
13) Evaluate: sin²73+ sin²47- sin73 sin47. 3/4
14) Show: 1+ cos56+ cos58 - cos66 = 4 cos28 cos29 sin33.
15) If sinx + sin2x= m and cosx + cos 2x = n, then show that (m²+ n²)(m²+ n² -3)= 2n.
16) If sec(x + a)+ sec(x - a) = 2secx, show that cosx =√2 cos(a/2).
17) If a= π/(2ⁿ +1), show that 2ⁿ cosa cos2a cos4a.....cos2ⁿ⁻¹a=1.
18) If tanx = a/b, show that, a cosec(x/3) - b sec(x/3)= 2√(a²+ b²).
19) Show that, cos2a = 2 sin²b+ 4 cos(a + b) sina sinx + cos2(a+ b).
20) If tan a/tanb = (1+ cos²a)/(1+ sin²a), show that, sin(3a+ b)= 7sin(a - b).
21) Show: sin²18+ sin² 24+sin²36+sin²42= 1+ sin²6+ sin²12.
22) Show that cot(15/2) = √2+ √3+ √4 + √6.
23) If cosx + cosy = a and sinx + siny= b, find sin(x + y) and cos(x - y) in terms of a, b. 2ab/(a²+ b²), (a²+ b²-2)/2
24) Evaluate: sin 36 sin 72 sin108 sin 144. 5/16
25) If tan(x/2)= √{(1- e)/(1+ e)} tan(y/2) show that cost = (cosx -e)/(1- e cosx).
26) If tan(x + y - z)/tan(x - y + z)= tany/tanz, show that sin(y - z)=0, or sin2x + sin2y + sin2z=0.
27) If a cosx + b sinx = c and b cosecx - a secx = c, show that, tan2x = 2ab/(a²- b²+ c²).
28) If (1+ √(1+ a))tanx = 1+ √(1- a), show that, sin4x= a.
29) show that: cosec (π/7) = cosec (2π/7) + cosec(3π/7).
30) Show: cos(π/32) = (1/2) √[2+ √{2+ √(2+ √2)}]. Hence or otherwise evaluate sin(π/32). (1/2) √[2- √{2+ √(2+ √2)}].
31) Show: cot70+ 4 cos70=√3.
32) show: 4 sin10+ √3 tan10= 1.
33) If tanx = (sina sinb)/(cosa+ cosb), show that one of the values of tan(x/2) is tan(a/2) tan(b/2)
34) if tan(A+ B)= 3 tanA, show sin(2A+ 2B)+ sin2A = 2 sin2B.
35) If tanx=√{(a-b)/(a+b)} tan(A/2), and cosy= (acos A +b)/(a+ bcosA), then show that, y= 2x.
36) If x=a(cosy+ siny sin 2y), and y= a(siny + cosy sin 2y), show that, (x+y)²⁾³+(x-y)²⁾³= 2a²⁾³
37) If cos a= cosx cosy, and cos b= cos m cosy, and tan(a/2)= tan(b/2)= tan(y/2) then prove sin²y= (secx -1)((sec m -1)
38) If tan(x+y), Tanz, tan(x-y) are in G. P. Prove that tan(x+z), tanx, tan(x-z) are also in G. P.
39) If tanx tany= √{(p-q)/(p+q)} prove (p -q cos 2x)(p-q cos 2y)= p² - q²
40) If xy+ y z+ z x= 1, show that x/(1-x²) + y/(1-y²) + z/(1-z²) = 4xyz/{(1-x²)(1-y²)(1-z²)}
41) If tanx= n(secx -1)², then prove that, cot³(x/2) - cot(x/2)= 2n
42) sin(π/14) sin(3π/14) sin(5π/14)= 1/8.
43) If x, y are two values of z satisfying atanz + b sec z= c, show tan(x+y)= 2ac/(a²- c²)
44) Show: tan 10+ tan 70 - tan 50=√3
45) If 3 sin²x + 2sin²y= 1 and 3sin 2x - 2 sin 2y=0, where x, y are positive acute angles, then prove that, x + 2y=π/2.
46) If cos³x/cos (y-3x) = sin³x/sin(y- 3x) = K, then show that, 2K² - K cosy -1= 0.
47) Show: cos(π/7) cos(3π/7)+ cos(3π/7) cos(5π/7)+ cos (5π/7) cos(π/7) = - 1/2
48) show: cos(π/11) + cos(3π/11)+ cos(5π/11) + cos(7π/11) + cos(9π/11) = 1/2.
49) If tan(x+y)= a+b and tan(x-y)= a - b, then show that, a tanx - b tany = a² - b².
50) If the equation acos 2x + b sin 2x= c has y and z as its solutions, Prove that,
A) tany+ tan z= 2b/(c+a)
B) tan y tanz= (c-a)/(c+a)
51) Show : sec²(π/16)+ sec²(3π/16)+ sec²(5π/16)+ sec²(7π/16)= 32 Hence, or otherwise find the value of tan²(π/16)+ tan²(3π/16) + tan²(5π/16)+ tan²(7π/16). 28
52) If {sin²(x+y)}/{sin²(z+y)}= sin 2x/sin 2z, show tanx tanz= tan²y
53) Find the range of the value of (3cosx + 4 sinx) sinx. Mx9/2, mn -1/2
54) Show: cos(3π/7) cos(4π/7) cos(6π/7)= 1/8.
55) Show: cos(2π/7) + cos(4π/7)+ cos(6π/7)= -1/2
56) sin(2π/7) + sin(4π/7) + sin(8π/7)= √7/2
