Evaluate

Evaluate

Let 0° < θ < 45°. Arrange
t₁ = (tanθ)^tanθ, t₂ = (tanθ)^cotθ
t₃ = (cotθ)^tanθ, t₄ = (cotθ)^cotθ
in decreasing order

a) t₄ > t₃ > t₂ > t₁

b) t₄ > t₃ > t₁ > t₂

c) t₃ > t₄ > t₂ > t₁

d) t₃ > t₄ > t₁ > t₂

The following composition of trigonometric buttons, such as sin, cos, tan, sin⁻¹, cos⁻¹, tan⁻¹, can be used to replace the broken reciprocal button on a calculator.

a) sin^{− 1} tan cos tan^{− 1}

b) tan^{− 1} sin tan^{− 1} cos

c) tan sin^{− 1} cos tan^{− 1}

d) cos^{− 1} sin cos^{− 1} tan

a)

b)

c)

d)

Let ABC be an acute angle triangle with area K. Solve
 

a)

b) a² + b² + c²

c)

d) 2 (a² + b² + c²)

Let a and b be real numbers in the interval . Prove that sin⁶a + cos⁶b + 3sin²acos²b = 1 if and only if a = b.

If sinx = ksin(A – x), show that
 

Express  in terms of tan3θ. Hence solve the equation
 

a)

b)

c)

d)

If cos⁻¹ + cos⁻¹y + cos⁻¹z = π, prove that
x² + y² + z² + 2xyz = .

a)

b)  1

c)

d) 1

If an angle α be divided into two parts, so that the ratio of the tangents of the parts is λ, the difference x between the parts is given by.

a)

b)

c)

d)

If tan (πcosθ) = cot (πsinθ), then

a)

b)

c)

d) ± 1

In any triangle ABC, show that
 

If the sides of a triangle are 51, 35 and 26 cm, find the sides of a triangle, on a base of 41cm, which shall have the same area and perimeter as the first.

a) 21 and 50

b) 20 and 51

c) 35 and 36

d) 26 and 45

a)

b)

c)

d)

a)

b)

c)

d)

a)

b)

c)

d)

Eliminate θ and ϕ from the relations
asin²θ + bcos²θ =1
acos²ϕ + bsin²ϕ=1
atanθ = btanϕ
and find the relation between a and b if 0 < b < 1, a > 1

a) a + b = ab

b)

c)

d)

For what values of  α and β does the equation
  follows from cos(2α + β) = 1

Find  if  and the angle α lies between 0° and 45°

a)

b)

c)

d)

The sum of three positive numbers α, β, γ is equal to  Compute the product cotα cotγ if we know that cotα, cotβ, cotγ form an arithmetic progression.

a) 1

b) 3

c) 5

d) 7

Prove that if α, β, γ are angles of a triangle, then
 

a)

b)

c)

d)

If α and β are angles in the first quadrant,
 then

a)

b)

c)

d)