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 is the only identity.
E) II, III, and IV are the only identities.

Question

Quizplus · Accepted Answer

To determine if the given equations are identities, we need to check whether they are true for all values of $	heta$.

I. $\csc(	heta)\sec(	heta)=\dfrac{1}{\sin(	heta)}\cdot\dfrac{1}{\cos(	heta)}=\dfrac{1}{\cos(	heta)\sin(	heta)}$ and $	an(	heta)=\dfrac{\sin(	heta)}{\cos(	heta)}$, so the given equation becomes $$\dfrac{1}{\cos(	heta)\sin(	heta)}=\dfrac{\sin(	heta)}{\cos(	heta)}.$$ Multiplying both sides by $\cos(	heta)\sin(	heta)$, we get $\sin^2(	heta)=\cos^2(	heta)\sin(	heta)$, which is not true for all values of $	heta$. Therefore, I is not an identity.

II. $\csc(	heta)	an(	heta)=\dfrac{1}{\sin(	heta)}\cdot\dfrac{\sin(	heta)}{\cos(	heta)}=\dfrac{1}{\cos(	heta)}=\sec(	heta)$, which is true for all values of $	heta$. Therefore, II is an identity.

III. $	an(	heta)\sec(	heta)=\dfrac{\sin(	heta)}{\cos(	heta)}\cdot\dfrac{1}{\cos(	heta)}=\dfrac{\sin(	heta)}{\cos^2(	heta)}$ and $\csc(	heta)=\dfrac{1}{\sin(	heta)}$, so the given equation becomes $$\dfrac{\sin(	heta)}{\cos^2(	heta)}=\dfrac{1}{\sin(	heta)}.$$ Multiplying both sides by $\cos^2(	heta)\sin(	heta)$, we get $\sin^2(	heta)=\cos^2(	heta)$, which is also not true for all values of $	heta$. Therefore, III is not an identity.

IV. $\csc(	heta)\sin(	heta)=\dfrac{1}{\sin(	heta)}\cdot\sin(	heta)=1$, which is true for all values of $	heta$. Therefore, IV is an identity.

Thus, the only identities are II and IV, so the answer is (B).

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 $\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )$