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

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