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

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 \[ 2.1 \quad \sin 68^{\circ} \cos 22^{\circ} + \cos 68^{\circ} \sin 22^{\circ} \] This expression can be rewritten using the sine addition formula: \[ \sin(a) \cos(b) + \cos(a) \sin(b) = \sin(a + b) \] Here, \(a = 68^{\circ}\) and \(b = 22^{\circ}\): \[ = \sin(68^{\circ} + 22^{\circ}) = \sin(90^{\circ}) = 1 \] ### Expression 2.2 \[ 2.2 \quad \cos 28^{\circ} \cos 2^{\circ} - \sin 28^{\circ} \sin 2^{\circ} \] This expression can be rewritten using the cosine subtraction formula: \[ \cos(a) \cos(b) - \sin(a) \sin(b) = \cos(a + b) \] Here, \(a = 28^{\circ}\) and \(b = 2^{\circ}\): \[ = \cos(28^{\circ} + 2^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \] ### Expression 2.3 \[ 2.3 \quad \cos 108^{\circ} \cos 188^{\circ} + \sin 252^{\circ} \sin 172^{\circ} \] This expression can be rewritten using the cosine addition formula: \[ \cos(a) \cos(b) + \sin(a) \sin(b) = \cos(a - b) \] Here, \(a = 108^{\circ}\) and \(b = 188^{\circ}\): \[ = \cos(108^{\circ} - 188^{\circ}) = \cos(-80^{\circ}) = \cos(80^{\circ}) = \sin(10^{\circ} \] ### Expression 2.4 \[ 2.4 \quad \sin 70^{\circ} \cos 10^{\circ} - \cos 80^{\circ} \cos 70^{\circ} \] We can rewrite \(\cos 80^{\circ}\) as \(\sin 10^{\circ}\): \[ = \sin 70^{\circ} \cos 10^{\circ} - \sin 10^{\circ} \cos 70^{\circ} \] Using the sine subtraction formula: \[ = \sin(70^{\circ} - 10^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \] ### Expression 2.5 \[ 2.5 \quad \sin 4x \sin 3x + \cos 4x \cos 3x \] This expression can be rewritten using the cosine addition formula: \[ \sin(a) \sin(b) + \cos(a) \cos(b) = \cos(a - b) \] Here, \(a = 4x\) and \(b = 3x\): \[ = \cos(4x - 3x) = \cos(x) \] ### Expression 2.6 \[ 2.6 \quad \sin(90^{\circ} - x) \cos x + \cos(90^{\circ} + x) \sin x \] Using the identities \(\sin(90^{\circ} - x) = \cos x\) and \(\cos(90^{\circ} + x) = -\sin x\): \[ = \cos x \cos x - \sin x \sin x = \cos^2 x - \sin^2 x = \cos(2x) \] ### Summary of Results 1. \(2.1 = 1\) 2. \(2.2 = \frac{\sqrt{3}}{2}\) 3. \(2.3 = \sin(10^{\circ})\) 4. \(2.4 = \frac{\sqrt{3}}{2}\) 5. \(2.5 = \cos(x)\) 6. \(2.6 = \cos(2x)\)