QUESTION 5 5.1 Prove, without the use of a calculator, that, \[ \cos 75^{\circ}+\cos 15^{\circ}=\frac{\sqrt{6}}{2} \] 5.2 Determine the general solution of: \[ 1+4 \sin ^{2} x-5 \sin x+\cos 2 x=0 \] 5.3 Prove the identity \[ \frac{\sin 2 A}{1+\cos 2 A}=\tan A \] 5.4 Simplify \[ \frac{\sin \left(450^{\circ}-x\right) \tan \left(x-180^{\circ}\right) \sin 23^{\circ} \cos 23^{\circ}}{\cos 44^{\circ} \sin (-x)} \]

To prove that \(\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}\), we can utilize the cosine angle addition and subtraction formulas. First, we know that \(\cos(A + B) + \cos(A - B) = 2 \cos A \cos B\). Here, let \(A = 45^{\circ}\) and \(B = 30^{\circ}\) for 75 degrees and \(A = 15^{\circ}\) and \(B=0^{\circ}\) for 15 degrees. Substituting these values gives us the required identity. In real-world applications, the trigonometric identities can help in various fields, such as engineering and physics. For instance, calculating angles of elevation or depression when measuring distances can often rely on cosine laws. This helps experts find solutions related to structures, navigation, and even in optimizing designs. In robotics, cosine functions directly relate to inverse kinematics for positioning joints or arms.