To find the value of [tex]\(\sec^2 30^\circ - \csc^2 60^\circ\)[/tex], we need to follow these steps:
1. Calculate [tex]\(\sec 30^\circ\)[/tex]:
- The secant function is the reciprocal of the cosine function.
- We know that [tex]\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)[/tex].
- Thus, [tex]\(\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\)[/tex].
2. Square the value of [tex]\(\sec 30^\circ\)[/tex]:
- [tex]\(\sec^2 30^\circ = \left(\frac{2\sqrt{3}}{3}\right)^2 = \frac{4 \cdot 3}{9} = \frac{12}{9} = \frac{4}{3}\)[/tex].
3. Calculate [tex]\(\csc 60^\circ\)[/tex]:
- The cosecant function is the reciprocal of the sine function.
- We know that [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex].
- Thus, [tex]\(\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\)[/tex].
4. Square the value of [tex]\(\csc 60^\circ\)[/tex]:
- [tex]\(\csc^2 60^\circ = \left(\frac{2\sqrt{3}}{3}\right)^2 = \frac{4 \cdot 3}{9} = \frac{12}{9} = \frac{4}{3}\)[/tex].
5. Subtract [tex]\(\csc^2 60^\circ\)[/tex] from [tex]\(\sec^2 30^\circ\)[/tex]:
- [tex]\(\sec^2 30^\circ - \csc^2 60^\circ = \frac{4}{3} - \frac{4}{3}\)[/tex].
6. Simplify the result:
- [tex]\(\frac{4}{3} - \frac{4}{3} = 0\)[/tex].
Therefore, the value of [tex]\(\sec^2 30^\circ - \csc^2 60^\circ\)[/tex] is [tex]\(-4.440892098500626 \times 10^{-16}\)[/tex]. This extremely small number is effectively [tex]\(0\)[/tex] considering the limits of numerical precision.
