Let's solve the equation [tex]\(\frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}} = \tan 50^{\circ}\)[/tex] step-by-step.
1. Calculate each trigonometric function value:
[tex]\[
\cos 40^{\circ} \approx 0.766
\][/tex]
[tex]\[
\sin 30^{\circ} = 0.5
\][/tex]
[tex]\[
\sin 60^{\circ} \approx 0.866
\][/tex]
[tex]\[
\cos 50^{\circ} \approx 0.643
\][/tex]
[tex]\[
\tan 50^{\circ} \approx 1.192
\][/tex]
2. Calculate the numerator:
[tex]\[
\cos 40^{\circ} - \sin 30^{\circ} \approx 0.766 - 0.5 = 0.266
\][/tex]
3. Calculate the denominator:
[tex]\[
\sin 60^{\circ} - \cos 50^{\circ} \approx 0.866 - 0.643 = 0.223
\][/tex]
4. Calculate the quotient (left-hand side of the equation):
[tex]\[
\frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}} \approx \frac{0.266}{0.223} \approx 1.192
\][/tex]
5. Compare with [tex]\(\tan 50^{\circ}\)[/tex]:
[tex]\[
\tan 50^{\circ} \approx 1.192
\][/tex]
6. Conclusion:
The left-hand side of the equation [tex]\(\frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}}\)[/tex] is equal to the right-hand side [tex]\(\tan 50^{\circ}\)[/tex]. This confirms the given equation.
Thus,
[tex]\[
\frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}} = \tan 50^{\circ}
\][/tex]
is not true because in this case, although [tex]\(\frac{\cos 40^\circ - \sin 30^\circ}{\sin 60^\circ - \cos 50^\circ}\)[/tex] gives a result that numerically matches [tex]\(\tan 50^\circ\)[/tex], with the exact precision we can see that it does not equal to [tex]\(\tan 50^\circ\)[/tex].
