To simplify the expression [tex]\(5 \csc \left(x - \frac{\pi}{2}\right)\)[/tex], we will use trigonometric identities.
1. Reciprocal Identity: The cosecant function is the reciprocal of the sine function. Therefore,
[tex]\[
\csc \left(x - \frac{\pi}{2}\right) = \frac{1}{\sin \left(x - \frac{\pi}{2}\right)}
\][/tex]
2. Sine Difference Identity: We can use the sine difference identity to simplify [tex]\(\sin \left(x - \frac{\pi}{2}\right)\)[/tex]:
[tex]\[
\sin \left(x - \frac{\pi}{2}\right) = \sin x \cos \left(\frac{\pi}{2}\right) - \cos x \sin \left(\frac{\pi}{2}\right)
\][/tex]
3. Substitute Known Values: We know that [tex]\(\cos \left(\frac{\pi}{2}\right) = 0\)[/tex] and [tex]\(\sin \left(\frac{\pi}{2}\right) = 1\)[/tex]. Substituting these values in:
[tex]\[
\sin \left(x - \frac{\pi}{2}\right) = \sin x \cdot 0 - \cos x \cdot 1 = -\cos x
\][/tex]
4. Substitute Back into Cosecant: Substitute [tex]\(\sin \left(x - \frac{\pi}{2}\right) = -\cos x\)[/tex] back into the reciprocal identity:
[tex]\[
\csc \left(x - \frac{\pi}{2}\right) = \frac{1}{-\cos x} = -\frac{1}{\cos x} = -\sec x
\][/tex]
5. Multiply by 5: Now, multiply by 5 as per the original expression:
[tex]\[
5 \csc \left(x - \frac{\pi}{2}\right) = 5 \left(- \sec x\right) = -5 \sec x
\][/tex]
Thus, the simplified expression is:
[tex]\[
5 \csc \left(x - \frac{\pi}{2}\right) = -\frac{5}{\cos x}
\][/tex]
