Option C : [tex]\sin \left(-\frac{\pi}{2}\right)[/tex] has the same value as [tex]\cos (3 \pi)[/tex]
Explanation:
It is given that to find the same value as [tex]\cos (3 \pi)[/tex] from the 4 options.
Now, we shall find the value of [tex]\cos (3 \pi)[/tex]
[tex]\cos (3 \pi)=\cos (2 \pi+\pi)[/tex]
[tex]\cos (3 \pi)=\cos (\pi)[/tex]
The value of [tex]\cos (\pi)[/tex] is -1.
Thus, the value of [tex]\cos (3 \pi)[/tex] is -1.
Option A : [tex]\sec \left(\frac{3 \pi}{2}\right)[/tex]
[tex]\sec \left(\frac{3 \pi}{2}\right)=\frac{1}{\cos \left(\frac{3 \pi}{2}\right)}[/tex]
Using the identity [tex]\cos (x)=\sin \left(\frac{\pi}{2}-x\right)[/tex],
[tex]\sec \left(\frac{3 \pi}{2}\right)=\frac{1}{\sin \left(\frac{\pi}{2}-\frac{3 \pi}{2}\right)}[/tex]
[tex]\sec \left(\frac{3 \pi}{2}\right)=\frac{1}{\sin \left(-\frac{2 \pi}{2}\right)}[/tex]
[tex]\sec \left(\frac{3 \pi}{2}\right)=\frac{1}{\sin \left(-{ \pi}\right)}[/tex]
Since, the value of [tex]\sin (\pi)[/tex] is 0. Thus, we have,
[tex]\sec \left(\frac{3 \pi}{2}\right)=\frac{1}{0\right)}[/tex]
Hence, the value of [tex]\sec \left(\frac{3 \pi}{2}\right)[/tex] is undefined.
Thus, the value of [tex]\sec \left(\frac{3 \pi}{2}\right)[/tex] is not the same as value of [tex]\cos (3 \pi)[/tex]
Therefore, Option A is not the correct answer.
Option B : [tex]\tan \left(\frac{\pi}{2}\right)[/tex]
The value of [tex]\tan \left(\frac{\pi}{2}\right)[/tex] is undefined.
Thus, the value of [tex]\tan \left(\frac{\pi}{2}\right)[/tex] is not the same as value of [tex]\cos (3 \pi)[/tex]
Therefore, Option B is not the correct answer.
Option C : [tex]\sin \left(\frac{-\pi}{2}\right)[/tex]
[tex]\sin \left(-\frac{\pi}{2}\right)=-\sin \left(\frac{\pi}{2}\right)[/tex]
Since, [tex]\sin \left(\frac{\pi}{2}\right)=1[/tex] Substituting, we have,
[tex]\sin \left(\frac{-\pi}{2}\right)=-1[/tex]
Thus, the value of [tex]\sin \left(\frac{-\pi}{2}\right)[/tex] is the same as the value of [tex]\cos (3 \pi)[/tex]
Therefore, Option C is the correct answer.
Option D : [tex]\cot (-\pi)[/tex]
Rewriting the angles,
[tex]\cot (-\pi)=\cot (-\pi+0\pi)[/tex]
Simplifying, we get,
[tex]\cot (-\pi)=\cot (0 \pi)[/tex]
The value of [tex]\cot (0 \pi)[/tex] is undefined.
Thus, the value of [tex]\cot (-\pi)[/tex] is not the same as the value of [tex]\cos (3 \pi)[/tex]
Therefore, Option D is not the correct answer.
