Use the graph to complete the identity.
-$\csc x + \tan ^ { 2 } x \csc x ; \cos x$ and $\sin x$
A)
$\frac { 1 } { \sin x \cos ^ { 2 } x }$
B) $\frac { \sin x + \cos x } { \sin x \cos x }$
C) $\frac { 1 } { \sin x \cos x }$
D) $\cos x - \sin x$

Question

Quizplus · Accepted Answer

The correct identity is derived by recognizing that $\csc x = \frac{1}{\sin x}$ and $\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$. Thus, $\csc x + \tan^2 x \csc x = \frac{1}{\sin x} + \frac{\sin^2 x}{\cos^2 x} \cdot \frac{1}{\sin x} = \frac{1 + \sin^2 x}{\sin x \cos^2 x}$. Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, we can simplify this to $\frac{1}{\sin x \cos^2 x}$, which matches choice A.

Use the Graph to Complete the Identity csc⁡x+tan⁡2xcsc⁡x;cos⁡x\csc x + \tan ^ { 2 } x \csc x ; \cos xcscx+tan2xcscx;cosx

Use the Graph to Complete the Identity $\csc x + \tan ^ { 2 } x \csc x ; \cos x$