Answer:
[tex]\boxed{d.\:\:\:\csc(x)=\sqrt{3}}[/tex]
Step-by-step explanation:
The given trigonometric equation is
[tex]\sin(x)+\cot(x) \cos(x)=\sqrt{3}[/tex]
Recall that;
[tex]\cot(x)=\frac{\cos(x)}{\sin(x)}[/tex]
This implies that;
[tex]\sin(x)+\frac{\cos(x)}{\sin(x)}\times \cos(x)=\sqrt{3}[/tex]
We collect LCM for the denominator on the left hand side to obtain;
[tex]\frac{\sin^2(x)+\cos^2(x)}{\sin(x)}=\sqrt{3}[/tex]
Recall that;
[tex]sin^2x+cos^2x=1[/tex]
[tex]\frac{1}{\sin(x)}=\sqrt{3}[/tex]
Recall again that;
[tex]\frac{1}{sinx}=cscx[/tex]
[tex]\Rightarrow \csc(x)=\sqrt{3}[/tex]
