Prove that

Let . For k = 1, 2, .  .  ., find  for all real numbers x.

a)

b)

c)

d)

In a triangle ABC, 3sinA + 4cosB = 6 and 4sinB + 3cosA = 1. Find the measure of angle C.

a)

b)

c)

d)

Let a, b, c, d be numbers in the interval [0, π] such that
sina + 7sinb = 4(sinc + 2sind)
cosa + 7cosb = 4(cosc + 2cosd)
Prove that 2cos(a – d) = 7cos(b – c)

(4cos²9° − 3) (4cos²27° − 3) =

a) sin9°

b) cos9°

c) tan9°

d) cot9°

Let ABC be a triangle.
 

a) 0

b) 1

c) tan²A + tan²B + tan²C

d) tanA tanB tanC

Let ABC be a triangle.
cos²A + cos²B + cos²C +2cosa cosB cosC

a) 0

b) 1

c) tan²A + tan²B + tan²C

d) tanA tanB tanC

Prove that  for all  where k ε Z

(1 + tan1°)(1 + tan2°).  .  .  . (1 + tan45°) =

a) 2²¹

b) 2²²

c) 2²³

d) 2⁴⁵

If  , sin²3α – sin²α =

a) sinα sin3α

b) cosα cos3α

c) sin2α sin3α

d) cos2α cos3α

If  , cosec2α + cosec4α =

a) sinα

b) cosecα

c) cosα

d) secα

If  , cosα is a root of

a) x³ - 4x² + 4x – 8  = 0

b) x³ + 4x² – 4x + 8  = 0

c) 8x³ - 4x² – 4x + 1  = 0

d) 8x³ + 4x² – 4x – 1  = 0

If  , cosα – cos2α + cos3α =

a)

b)

c)

d) 1

Prove that  Hence or otherwise evaluate the integral .

If the expression  is real then the set of all possible values of x is .  .  .  .

a) x = 2nπ or mπ + π/4

b) x = nπ or mπ + π/4

c) x = 2nπ or 2mπ + π/4

d) x = nπ or 2mπ + π/4

Does the equation  |sinx| = sinx + 3 have any roots?

a) No roots

b) One root

c) Two roots

d) ∞ roots

Does the equation |tanx| = tanx + 3 have any roots in [0, 2π]?

a) No roots

b) Has roots in

c) Has roots in 

d) Has roots in

The value of cot  is

a)

b)

c)

d)

If sin^-1  + cosec^-1  =  

Then the value of x is

a) 1

b) 3

c) 4

d) 5

The number of values of x in the interval [0, 3π] satisfying the equation 2sin²x + 5sinx – 3 = 0 is

a) 6

b) 1

c) 2

d) 4

If 0 < x < π and cos x + sin x =  then tan x is

a)

b)

c)

d)

In a triangle PQR, angle R = . If tan  and tan  are roots of ax² + bx + c = 0, a ≠ 0 then

a) b =a + c

b) b = c

c) c = a + b

d) a = b + c

In a triangle ABC, let angle C = . If r is the inradius and R is the circumradius of the triangle ABC then 2(r + R) equals

a) c + a

b) a + b + c

c) a + b

d) b + c