Sure, let's solve this step-by-step:
Given expression:
[tex]\[ \cos 412^\circ - \sin^4 \left(\frac{412^\circ}{2}\right) = \cos 24^\circ \][/tex]
Let's find each term individually.
1. Reduce the angle [tex]\( 412^\circ \)[/tex] to a value within the range [tex]\([0^\circ, 360^\circ)\)[/tex]:
We can achieve this by calculating [tex]\( 412 \mod 360 \)[/tex]:
[tex]\[ 412 \mod 360 = 52^\circ \][/tex]
So, we need to work with [tex]\( 52^\circ \)[/tex].
2. Calculate [tex]\( \cos 52^\circ \)[/tex]:
Using the cosine function:
[tex]\[ \cos 52^\circ \approx 0.6156614753256583 \][/tex]
3. Calculate [tex]\( \sin \left(\frac{412^\circ}{2}\right) \)[/tex]:
First, calculate the angle:
[tex]\[ \frac{412^\circ}{2} = 206^\circ \][/tex]
Now, evaluate the sine function:
[tex]\[ \sin 206^\circ \approx 0.4383711467890774 \][/tex]
4. Evaluate [tex]\( \sin^4 206^\circ \)[/tex]:
Raise the sine value to the power of 4:
[tex]\[ (0.4383711467890774)^4 \approx 0.0371329163872124 \][/tex]
5. Calculate [tex]\( \cos 24^\circ \)[/tex]:
Using the cosine function:
[tex]\[ \cos 24^\circ \approx 0.9135454576426009 \][/tex]
6. Form the original expression:
Substitute the values we found into the expression:
[tex]\[ \cos 52^\circ - (\sin 206^\circ)^4 \][/tex]
[tex]\[ 0.6156614753256583 - 0.0371329163872124 \approx 0.5785285589384459 \][/tex]
7. Compare with [tex]\( \cos 24^\circ \)[/tex]:
[tex]\[ \cos 24^\circ \approx 0.9135454576426009 \][/tex]
7. Final Comparison:
The left-hand side:
[tex]\[0.5785285589384459 \][/tex]
is not equal to the right-hand side:
[tex]\[0.9135454576426009 \][/tex]
Thus, upon calculation, we can deduce that the given equation [tex]\( \cos 412^\circ - \sin^4 \left(\frac{412^\circ}{2}\right) = \cos 24^\circ \)[/tex] does not seem to hold true based on the numerical values calculated.
