Trigonometric identities

Prove that (cos θ + sin θ)² ≡ 1 + sin 2θ.

Show that the equation 2 cot²x - 9 cosec x - 3 = 0 can be expressed in the form

In this question you must show detailed reasoning.

Hence solve, for 0 < θ < π,

Give your answers correct to 3 decimal places.

The small angle approximation for sin2 θ is used to find an approximation for the smallest positive solution of the equation 2 cot²2θ - 9 cosec 2θ - 3 = 0.

Show that this approximate solution is accurate to 2 decimal places.

The curve C is defined, for 0 ⩽ t < 2 π, by the parametric equations

x = 4 k + k sin t, y = 2 + 4 cos t,

where k is a constant.

Find a cartesian equation for C. You do not need to simplify your answer.

You are given that C is a circle.

Determine the radius of C.

Find the possible coordinates for the centre of C.

Use the result cos (A + B) = cos A cos B - sin A sin B to show that

cos (A - B) = cos A cos B + sin A sin B.

The function f (θ) is defined as cos (θ + 30 °) cos (θ - 30 °), where θ is in degrees.

Determine the following.

• The maximum value of f (θ)

• The smallest positive value of θ for which this maximum value occurs

Determine the following.

• The minimum value of f (θ)

• The smallest positive value of θ for which this minimum value occurs

In this question you must show detailed reasoning.

Show that the equation m sec θ + 3 cos θ = 4 sin θ can be expressed in the form

m tan²θ - 4 tan θ + (m + 3) = 0.

It is given that there is only one value of θ, for 0 < θ < π, satisfying the equation m sec θ + 3 cos θ = 4 sin θ.

Given also that m is a negative integer, find this value of θ, correct to 3 significant figures.

Functions f and g are defined for 0 ⩽ x ⩽ 2 π by f (x) = 2 tan x and g (x) = sec x.

State the range of f.

State the range of g.

Show that fg (0.6) = 5.33, correct to 3 significant figures.

Explain why f^-1g(0.6) is not defined.