Rewrite the expressions that follow as the sine or cosine of a single angle and evaluate where possible without a calculator: \( 2.1 \sin 68^{\circ} \cos 22^{\circ} + \cos 68^{\circ} \sin 22^{\circ} \) \( 2.2 \cos 28^{\circ} \cos 2^{\circ} - \sin 28^{\circ} \sin 2^{\circ} \) \( 2.3 \cos 108^{\circ} \cos 188^{\circ} + \sin 252^{\circ} \sin 172^{\circ} \) \( 2.4 \sin 70^{\circ} \cos 10^{\circ} - \cos 80^{\circ} \cos 70^{\circ} \) \( 2.5 \sin 4 x \sin 3 x + \cos 4 x \cos 3 x \) \( 2.6 \sin \left(90^{\circ}-x\right) \cos x + \cos \left(90^{\circ}+x\right) \sin x \)

To rewrite the given expressions as the sine or cosine of a single angle, we can use the angle addition and subtraction formulas. Let's go through each expression step by step. ### Expression 2.1 **Expression:** \[ 2.1 \sin 68^{\circ} \cos 22^{\circ} + \cos 68^{\circ} \sin 22^{\circ} \] **Using the sine addition formula:** \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] Here, \(a = 68^{\circ}\) and \(b = 22^{\circ}\). **Rewriting:** \[ 2.1 = \sin(68^{\circ} + 22^{\circ}) = \sin(90^{\circ}) = 1 \] ### Expression 2.2 **Expression:** \[ 2.2 \cos 28^{\circ} \cos 2^{\circ} - \sin 28^{\circ} \sin 2^{\circ} \] **Using the cosine subtraction formula:** \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] Here, \(a = 28^{\circ}\) and \(b = 2^{\circ}\). **Rewriting:** \[ 2.2 = \cos(28^{\circ} + 2^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \] ### Expression 2.3 **Expression:** \[ 2.3 \cos 108^{\circ} \cos 188^{\circ} + \sin 252^{\circ} \sin 172^{\circ} \] **Using the cosine addition formula:** \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] Here, \(a = 252^{\circ}\) and \(b = 172^{\circ}\). **Rewriting:** \[ 2.3 = \cos(252^{\circ} - 172^{\circ}) = \cos(80^{\circ}) \] ### Expression 2.4 **Expression:** \[ 2.4 \sin 70^{\circ} \cos 10^{\circ} - \cos 80^{\circ} \cos 70^{\circ} \] **Using the sine subtraction formula:** \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] Here, \(a = 70^{\circ}\) and \(b = 10^{\circ}\). **Rewriting:** \[ 2.4 = \sin(70^{\circ} - 10^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \] ### Expression 2.5 **Expression:** \[ 2.5 \sin 4x \sin 3x + \cos 4x \cos 3x \] **Using the cosine addition formula:** \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] Here, \(a = 4x\) and \(b = 3x\). **Rewriting:** \[ 2.5 = \cos(4x - 3x) = \cos(x) \] ### Expression 2.6 **Expression:** \[ 2.6 \sin(90^{\circ}-x) \cos x + \cos(90^{\circ}+x) \sin x \] **Using the sine addition formula:** \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] Here, \(a = 90^{\circ} - x\) and \(b = x\). **Rewriting:** \[ 2.6 = \sin(90^{\circ} - x + x) = \sin(90^{\circ}) = 1 \] ### Summary of Results - \(2.1 = 1\) - \(2.2 = \frac{\sqrt{3}}{2}\) - \(2.3 = \cos(80^{\circ})\) - \(2.4 = \frac{\sqrt{3}}{2}\) - \(2.5 = \cos(x)\) - \(2.6 = 1\)