Certainly! Let’s start by thoroughly breaking down and analyzing the given problem.
We are asked to prove that [tex]\(\frac{\cos 14^\circ - \sin 14^\circ}{\cos 4^\circ + \sin 14^\circ} = \cot 59^\circ\)[/tex].
First, let’s evaluate the trigonometric values needed for our expression.
1. Calculate [tex]\(\cos 14^\circ\)[/tex]:
[tex]\[ \cos 14^\circ \approx 0.9703 \][/tex]
2. Calculate [tex]\(\sin 14^\circ\)[/tex]:
[tex]\[ \sin 14^\circ \approx 0.2419 \][/tex]
3. Calculate [tex]\(\cos 4^\circ\)[/tex]:
[tex]\[ \cos 4^\circ \approx 0.9976 \][/tex]
4. Calculate [tex]\(\sin 4^\circ\)[/tex]:
[tex]\[ \sin 4^\circ \approx 0.0698 \][/tex]
5. Calculate [tex]\(\cot 59^\circ\)[/tex]:
[tex]\[ \cot 59^\circ = \frac{1}{\tan 59^\circ} \approx 0.6009 \][/tex]
Next, let's insert these values into our original left-hand side expression:
[tex]\[ \text{LHS} = \frac{\cos 14^\circ - \sin 14^\circ}{\cos 4^\circ + \sin 14^\circ} \][/tex]
Plugging in the values we have:
[tex]\[ \text{LHS} = \frac{0.9703 - 0.2419}{0.9976 + 0.2419} \][/tex]
Let’s compute the numerator:
[tex]\[ 0.9703 - 0.2419 \approx 0.7284 \][/tex]
And the denominator:
[tex]\[ 0.9976 + 0.2419 \approx 1.2395 \][/tex]
Therefore, the left-hand side becomes:
[tex]\[ \text{LHS} = \frac{0.7284}{1.2395} \approx 0.5876 \][/tex]
Thus, the computed value for the left-hand side expression is approximately:
[tex]\[ 0.5876 \][/tex]
Now, let's recall our computed value for [tex]\(\cot 59^\circ\)[/tex]:
[tex]\[ \cot 59^\circ \approx 0.6009 \][/tex]
Finally, we compare the left-hand side and right-hand side:
[tex]\[ \frac{\cos 14^\circ - \sin 14^\circ}{\cos 4^\circ + \sin 14^\circ} \approx 0.5876 \][/tex]
[tex]\[ \cot 59^\circ \approx 0.6009 \][/tex]
These two values are not equal, thus:
[tex]\[ \frac{\cos 14^\circ - \sin 14^\circ}{\cos 4^\circ + \sin 14^\circ} \neq \cot 59^\circ \][/tex]
Therefore, we conclude that the given identity is not true based on the calculated trigonometric approximations. The initial hypothesis fails since:
[tex]\[ \boxed{\frac{\cos 14^\circ - \sin 14^\circ}{\cos 4^\circ + \sin 14^\circ} \neq \cot 59^\circ} \][/tex]
