Answer:
[tex] cot x = \frac{cos x}{sin x}[/tex]
[tex] cos x \frac{cos x}{sin x} + sin x[/tex]
[tex] \frac{cos^2 x}{sin x} +sin x[/tex]
[tex] sin^2 x + cos^2 x =1 [/tex]
Solving for [tex]cos^2 x[/tex] we got [tex] cos^2 x =1 -sin^2 x[/tex] and replacing this we got:
[tex] \frac{1-sin^2 x}{sin x} +sin x [/tex]
[tex] \frac{1}{sin x} -\frac{sin^2 x}{sin x} +sin x[/tex]
[tex] csc x -sin x + sin x = csc x[/tex]
And then the best option for this case would be:
b.csc x
Step-by-step explanation:
For this case we have the following expression given:
[tex] cos x cot x + sin x [/tex]
We know from math properties that the definition for cot is [tex] cot x = \frac{cos x}{sin x}[/tex]
If we use this definition we got:
[tex] cos x \frac{cos x}{sin x} + sin x[/tex]
[tex] \frac{cos^2 x}{sin x} +sin x[/tex]
Now we can use the following identity:
[tex] sin^2 x + cos^2 x =1 [/tex]
Solving for [tex]cos^2 x[/tex] we got [tex] cos^2 x =1 -sin^2 x[/tex] and replacing this we got:
[tex] \frac{1-sin^2 x}{sin x} +sin x [/tex]
[tex] \frac{1}{sin x} -\frac{sin^2 x}{sin x} +sin x[/tex]
[tex] csc x -sin x + sin x = csc x[/tex]
And then the best option for this case would be:
b.csc x
