Let's find the result of [tex]\(\left(\tan 72^{\circ}\right) \cdot \left(\cos 72^{\circ}\right)\)[/tex].
### Step-by-Step Solution:
1. Convert degrees to radians:
First, we need to convert the angle from degrees to radians because trigonometric functions often work with radian measure.
Given angle [tex]\(\theta = 72^\circ\)[/tex],
[tex]\[
\theta \text{ in radians} = \frac{72^\circ \times \pi}{180^\circ} = \frac{72\pi}{180} = \frac{2\pi}{5} \approx 1.2566370614 \text{ radians}
\][/tex]
2. Calculate [tex]\(\tan 72^\circ\)[/tex]:
The value of [tex]\(\tan(72^\circ)\)[/tex] is:
[tex]\[
\tan 72^\circ \approx 3.0776835371752527
\][/tex]
3. Calculate [tex]\(\cos 72^\circ\)[/tex]:
The value of [tex]\(\cos(72^\circ)\)[/tex] is:
[tex]\[
\cos 72^\circ \approx 0.30901699437494745
\][/tex]
4. Calculate the product:
Now, we calculate the product [tex]\(\left(\tan 72^\circ\right) \cdot \left(\cos 72^\circ\right)\)[/tex]:
[tex]\[
(\tan 72^\circ) \cdot (\cos 72^\circ) \approx 3.0776835371752527 \times 0.30901699437494745
\][/tex]
When we multiply these values, we get:
[tex]\[
(\tan 72^\circ) \cdot (\cos 72^\circ) \approx 0.9510565162951534
\][/tex]
5. Interpret the result:
Notice that the expression [tex]\(\left(\tan 72^\circ\right) \cdot \left(\cos 72^\circ\right)\)[/tex] simplifies to [tex]\(\sin 72^\circ\)[/tex] because of the trigonometric identity:
[tex]\[
\tan(\theta) \cdot \cos(\theta) = \sin(\theta) \quad \text{for any } \theta
\][/tex]
Therefore, the product [tex]\(\left(\tan 72^\circ\right) \cdot \left(\cos 72^\circ\right)\)[/tex] is indeed:
[tex]\[
\sin 72^\circ \approx 0.9510565162951534
\][/tex]
Thus, we have verified that:
[tex]\[
(\tan 72^\circ) \cdot (\cos 72^\circ) = \sin 72^\circ \approx 0.9510565162951534
\][/tex]
