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

Let's rewrite and evaluate each expression step-by-step. **2.1** Using the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \): \[ 2.1 = \sin(68^{\circ} + 22^{\circ}) = \sin(90^{\circ}) = 1 \] **2.2** Using the cosine subtraction formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \): \[ 2.2 = \cos(28^{\circ} - 2^{\circ}) = \cos(26^{\circ}) \] **2.3** Using the cosine addition formula \( \cos(a + b) = \cos a \cos b - \sin a \sin b \): \[ 2.3 = \cos(108^{\circ} - 188^{\circ}) = \cos(-80^{\circ}) = \cos(80^{\circ}) \] **2.4** Using the cosine subtraction formula: \[ 2.4 = \sin(70^{\circ}) \cos(10^{\circ}) - \cos(80^{\circ}) \cos(70^{\circ}) = \sin(70^{\circ} - 10^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \] **2.5** Using the cosine addition formula: \[ 2.5 = \cos(4x - 3x) = \cos(x) \] **2.6** Using the identity \( \sin(90^{\circ} - x) = \cos x \) and \( \cos(90^{\circ} + x) = -\sin x \): \[ 2.6 = \cos x \cos x - (-\sin x) \sin x = \cos^2 x + \sin^2 x = 1 \] These rewritten expressions give us a concise way of evaluating the trigonometric combinations without a calculator!