Determine which of the following are trigonometric identities. I. $\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )$
II. $\csc ( \theta ) \tan ( \theta ) = \sec ( \theta )$
III. $\tan ( \theta ) \sec ( \theta ) = \csc ( \theta )$
IV. $\csc ( \theta ) \sin ( \theta ) = 1$
A) II is the only identity.
B) II and IV are the only identities.
C) I, III, and IV are the only identities.
D) I the only identity.
E) II, III, and IV are the only identities.

Question

Quizplus · Accepted Answer

To determine if an equation is a trigonometric identity, we need to verify if the equation holds true for all possible values of the variable $	heta$. In this case, we can use algebraic manipulation and recognize some Pythagorean identities to simplify each equation:
I. $\csc(	heta)\sec(	heta) = \frac{1}{\sin(	heta)} \cdot \frac{1}{\cos(	heta)} = \frac{\cos(	heta)}{\sin(	heta)\cos(	heta)} = \frac{\cos(	heta)}{\sin(	heta)} = \frac{\sin(	heta)}{\cos(	heta)} \cdot \frac{\cos(	heta)}{\sin(	heta)} = 	an(	heta)$
II. $\csc(	heta)	an(	heta) = \frac{1}{\sin(	heta)} \cdot \frac{\sin(	heta)}{\cos(	heta)} = \frac{1}{\cos(	heta)} = \sec(	heta)$
III. $	an(	heta)\sec(	heta) = \frac{\sin(	heta)}{\cos(	heta)} \cdot \frac{1}{\cos(	heta)} = \frac{\sin(	heta)}{\cos^2(	heta)} = \frac{1}{\cos(	heta)} \cdot \frac{1}{\sin(	heta)} = \csc(	heta)$
IV. $\csc(	heta)\sin(	heta) = \frac{1}{\sin(	heta)} \cdot \sin(	heta) = 1$
Therefore, II and IV are the only identities.

Determine Which of the Following Are Trigonometric Identities csc⁡(θ)sec⁡(θ)=tan⁡(θ)\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )csc(θ)sec(θ)=tan(θ)

Determine Which of the Following Are Trigonometric Identities $\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )$