To solve the expression [tex]\(\left(\sqrt{1-\sin^2 100^\circ}\right) \sec 100^\circ\)[/tex], we'll break it down step-by-step.
1. Calculate [tex]\(\sin^2 100^\circ\)[/tex]:
[tex]\[
\sin^2 100^\circ \approx 0.9698463103929541
\][/tex]
2. Find [tex]\(1 - \sin^2 100^\circ\)[/tex]:
[tex]\[
1 - \sin^2 100^\circ = 1 - 0.9698463103929541 \approx 0.0301536896070459
\][/tex]
3. Calculate the square root of [tex]\(1 - \sin^2 100^\circ\)[/tex]:
[tex]\[
\sqrt{1 - \sin^2 100^\circ} = \sqrt{0.0301536896070459} \approx 0.1736481776669306
\][/tex]
4. Find [tex]\(\sec 100^\circ\)[/tex] (which is [tex]\( \frac{1}{\cos 100^\circ} \)[/tex]):
[tex]\[
\sec 100^\circ \approx -5.758770483143635
\][/tex]
5. Now multiply [tex]\(\sqrt{1 - \sin^2 100^\circ}\)[/tex] by [tex]\(\sec 100^\circ\)[/tex]:
[tex]\[
\left(\sqrt{1 - \sin^2 100^\circ}\right) \sec 100^\circ = 0.1736481776669306 \times -5.758770483143635 \approx -1.0000000000000018
\][/tex]
Thus, the final result is:
[tex]\[
\left(\sqrt{1-\sin^2 100^\circ}\right) \sec 100^\circ \approx -1.0000000000000018
\][/tex]
