Determine which of the following are trigonometric identities.
I. $\cos ^ { 4 } ( \mathrm { t } ) + \sin ^ { 4 } ( \mathrm { t } ) = 1 - 2 \sin ^ { 2 } ( \mathrm { t } ) + 2 \sin ^ { 4 } ( \mathrm { t } )$
II. $\sin ^ { 5 } ( \mathrm { t } ) = \sin ^ { 3 } ( \mathrm { t } ) \cos ^ { 2 } ( \mathrm { t } ) - \sin ^ { 3 } ( \mathrm { t } )$
III. $\sin ^ { 3 } ( \mathrm { t } ) \cos ^ { 2 } ( \mathrm { t } ) = \left( \cos ^ { 2 } ( \mathrm { t } ) - \cos ^ { 4 } ( \mathrm { t } ) \right) \sin ( \mathrm { t } )$
A) I is the only identity.
B) I and III are the only identities.
C) III is the only identity.
D) I, II, and III are identities.
E) I and II are the only identities.

Question

Quizplus · Accepted Answer

I is a trigonometric identity derived from expanding $(\cos^2(t) + \sin^2(t))^2 = 1^2$. III can be derived using the identity $\sin^2(t) + \cos^2(t) = 1$ and manipulating it. II is not an identity because it incorrectly relates powers of sine and cosine without accounting for their fundamental relationship.

Determine Which of the Following Are Trigonometric Identities cos⁡4(t)+sin⁡4(t)=1−2sin⁡2(t)+2sin⁡4(t)\cos ^ { 4 } ( \mathrm { t } ) + \sin ^ { 4 } ( \mathrm { t } ) = 1 - 2 \sin ^ { 2 } ( \mathrm { t } ) + 2 \sin ^ { 4 } ( \mathrm { t } )cos4(t)+sin4(t)=1−2sin2(t)+2sin4(t)

Determine Which of the Following Are Trigonometric Identities $\cos ^ { 4 } ( \mathrm { t } ) + \sin ^ { 4 } ( \mathrm { t } ) = 1 - 2 \sin ^ { 2 } ( \mathrm { t } ) + 2 \sin ^ { 4 } ( \mathrm { t } )$