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. $\cos^4(t) + \sin^4(t) = ( \cos^2(t) + \sin^2(t))^2 - 2\cos^2(t)\sin^2(t) = 1-2\cos^2(t)\sin^2(t)+2\sin^4(t)$, which is an identity.

II. Using the identity $\sin^2(t) + \cos^2(t) = 1$, we can write $\sin^5(t) = \sin^3(t)\sin^2(t) = \sin^3(t)(1-\cos^2(t)) = \sin^3(t)\cos^2(t) - \sin^3(t)$. Therefore, this is not an identity.

III. $\sin^3(t)\cos^2(t) = \sin(t)\sin^2(t)\cos^2(t) = \sin(t)\cos^2(t)(1-\sin^2(t)) = \sin(t)(\cos^2(t)-\cos^4(t))$, which is an identity.

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 $\cos ^ { 4 } ( \mathrm { t } ) + \sin ^ { 4 } ( \mathrm { t } ) = 1 - 2 \sin ^ { 2 } ( \mathrm { t } ) + 2 \sin ^ { 4 } ( \mathrm { t } )$