Simplify the following without the use of a calculator: \( \frac{\cos \left(40^{\circ}-x\right) \cos x-\sin \left(40^{\circ}-x\right) \sin x}{\sin 205^{\circ} \cos 155^{\circ}} \) Calculate the value of \( \frac{\cos 2^{\circ}+\sqrt{3} \sin \left(-2^{\circ}\right)}{4 \sin 346^{\circ} \cos 194^{\circ}} \) without the use of a calculator.
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To simplify the fraction \( \frac{\cos \left(40^{\circ}-x\right) \cos x-\sin \left(40^{\circ}-x\right) \sin x}{\sin 205^{\circ} \cos 155^{\circ}} \), we can employ the cosine angle subtraction formula: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B. \] In our case, \( A = 40^{\circ} \) and \( B = x \). Thus, \[ \cos(40^{\circ} - x) \cos x - \sin(40^{\circ} - x) \sin x = \cos(40^{\circ} + x). \] Now, for the denominator: \[ \sin 205^{\circ} = \sin(180^{\circ} + 25^{\circ}) = -\sin 25^{\circ}, \] and \[ \cos 155^{\circ} = \cos(180^{\circ} - 25^{\circ}) = -\cos 25^{\circ}. \] Thus, \[ \sin 205^{\circ} \cos 155^{\circ} = (-\sin 25^{\circ})(-\cos 25^{\circ}) = \sin 25^{\circ} \cos 25^{\circ} = \frac{1}{2} \sin(50^{\circ}). \] Now, substituting back, the expression simplifies to: \[ \frac{\cos(40^{\circ} + x)}{\frac{1}{2} \sin(50^{\circ})} = \frac{2 \cos(40^{\circ} + x)}{\sin(50^{\circ})}. \] We know that \( \sin(50^{\circ}) = \cos(40^{\circ}) \), thus finally, the entire expression simplifies to: \[ \frac{2 \cos(40^{\circ} + x)}{\cos(40^{\circ})}. \] Next, we calculate \( \frac{\cos 2^{\circ} + \sqrt{3} \sin(-2^{\circ})}{4 \sin 346^{\circ} \cos 194^{\circ}} \). First, let’s simplify the numerator: \[ \sin(-2^{\circ}) = -\sin 2^{\circ}, \] Thus, \[ \cos 2^{\circ} + \sqrt{3} \sin(-2^{\circ}) = \cos 2^{\circ} - \sqrt{3}\sin 2^{\circ}. \] For the denominator: \[ \sin 346^{\circ} = \sin(360^{\circ} - 14^{\circ}) = -\sin 14^{\circ}, \] and \[ \cos 194^{\circ} = \cos(180^{\circ} + 14^{\circ}) = -\cos 14^{\circ}. \] Thus, \[ 4 \sin 346^{\circ} \cos 194^{\circ} = 4 (-\sin 14^{\circ})(-\cos 14^{\circ}) = 4 \sin 14^{\circ} \cos 14^{\circ} = 2 \sin 28^{\circ}. \] Finally, putting it all together, we have: \[ \frac{\cos 2^{\circ} - \sqrt{3} \sin 2^{\circ}}{2 \sin 28^{\circ}}. \] We can further simplify if needed, but this represents the expression without a calculator.
