To determine which expression is a trigonometric identity, we need to evaluate the options given and see which one is a well-known true trigonometric relationship.
1. Option 1: [tex]\(\tan ^2 u+\cot ^2 u=1\)[/tex]
- This is not a correct trigonometric identity. The correct relationship between [tex]\(\tan u\)[/tex] and [tex]\(\cot u\)[/tex] is:
[tex]\[
\tan u = \frac{1}{\cot u}
\][/tex]
- Additionally, there is a Pythagorean identity:
[tex]\[
\tan^2 u + 1 = \sec^2 u
\][/tex]
2. Option 2: [tex]\(\sec u=\sin \frac{1}{u}\)[/tex]
- This is not a correct identity. The secant function is defined as:
[tex]\[
\sec u = \frac{1}{\cos u}
\][/tex]
- There is no standard trigonometric identity involving [tex]\(\sec u\)[/tex] and [tex]\(\sin \frac{1}{u}\)[/tex] in this form.
3. Option 3: [tex]\(\sin \left(\frac{\pi}{2}+u\right)=\sin u\)[/tex]
- This is also not correct. According to the sine addition formula, we have:
[tex]\[
\sin \left(\frac{\pi}{2}+u\right) = \cos u
\][/tex]
4. Option 4: [tex]\(\sec u=\frac{1}{\cos u}\)[/tex]
- This is a correct trigonometric identity. The secant function is defined as the reciprocal of the cosine function:
[tex]\[
\sec u = \frac{1}{\cos u}
\][/tex]
Given the options, the trigonometric identity that is true is:
[tex]\(\sec u=\frac{1}{\cos u}\)[/tex]
So, the correct identity is found in option 4.
