Answer:
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)=0.73$[/tex]
Step-by-step explanation:
The identity you will use is:
[tex]$\csc \left(x\right)=\frac{1}{\sin \left(x\right)}$[/tex]
So,
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)$[/tex]
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)=\frac{1}{\sin \left(-\frac{\pi }{2}+\theta\right)}$[/tex]
Now, using the difference of sin
Note: state that [tex]\text{sin}(\alpha\pm \beta)=\text{sin}(\alpha) \text{cos}(\beta) \pm \text{cos}(\alpha) \text{sin}(\beta)[/tex]
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)=\frac{1}{-\cos \left(\theta\right)\sin \left(\frac{\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)\sin \left(\theta\right)}$[/tex]
Solving the difference of sin:
[tex]$-\cos \left(\theta\right)\sin \left(\frac{\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)\sin \left(\theta\right)$[/tex]
[tex]-\cos \left(\theta\right) \cdot 1+0\cdot \sin \left(\theta\right)[/tex]
[tex]-\text{cos} \left(\theta\right)[/tex]
Then,
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)=-\frac{1}{\cos \left(\theta\right)}$[/tex]
Once
[tex]\text{sec}(-\theta)=\text{sec}(\theta)[/tex]
And, [tex]\text{sec}(\theta)=-0.73[/tex]
[tex]$-\frac{1}{\cos \left(\theta\right)}=-\text{sec}(\theta)$[/tex]
[tex]$-\frac{1}{\cos \left(\theta\right)}=-(-0.73)$[/tex]
[tex]$-\frac{1}{\cos \left(\theta\right)}=0.73$[/tex]
Therefore,
[tex]$\csc \left(\theta-\frac{\pi }{2}\right)=0.73$[/tex]
