Trigonometric proof

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

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.

Hence solve, for 0 < θ < 2π, 3(cot θ + cosec θ)² = 2 sec θ.