Q: What is the Pythagorean identity?

$$\sin^2 \theta + \cos^2 \theta = 1.$$

Q: What is ratio identities?

$$\tan \theta = \frac{\sin \theta}{\cos \theta}, \qquad \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}.$$

Q: What is complementary angle identities?

The complement of $\theta$ is $\frac{\pi}{2} - \theta$ (in degrees, $90^\circ - \theta$). Co-function pairs:

Q: What is supplementary, negative angle, and reflection identities?

$$\sin(\pi - \theta) = \sin \theta, \qquad \cos(\pi - \theta) = -\cos \theta, \qquad \tan(\pi - \theta) = -\tan \theta.$$

Q: What is double angle identities?

$$\sin 2\theta = 2 \sin \theta \cos \theta.$$

Q: What is simplify using Pythagoras?

Simplify $\frac{1 - \cos^2 \theta}{\sin \theta}$.

Q: What is prove an identity?

Prove $\sec^2 \theta - 1 = \tan^2 \theta$.

Q: What is treating $\sin 2\theta$ as $2 \sin \theta$?

$\sin 2\theta = 2 \sin \theta \cos \theta$. The factor $\cos \theta$ is essential.

Question 1

What is the Pythagorean identity?

Accepted Answer

$$\sin^2 	heta + \cos^2 	heta = 1.$$

Question 2

What is ratio identities?

Accepted Answer

$$	an 	heta = \frac{\sin 	heta}{\cos 	heta}, \qquad \cot 	heta = \frac{\cos 	heta}{\sin 	heta} = \frac{1}{	an 	heta}.$$

Question 3

What is complementary angle identities?

Accepted Answer

The complement of $	heta$ is $\frac{\pi}{2} - 	heta$ (in degrees, $90^\circ - 	heta$). Co-function pairs:

Question 4

What is supplementary, negative angle, and reflection identities?

Accepted Answer

$$\sin(\pi - 	heta) = \sin 	heta, \qquad \cos(\pi - 	heta) = -\cos 	heta, \qquad 	an(\pi - 	heta) = -	an 	heta.$$

Question 5

What is double angle identities?

Accepted Answer

$$\sin 2	heta = 2 \sin 	heta \cos 	heta.$$

Question 6

What is power reduction (useful when integrating)?

Accepted Answer

From the double angle identities,

Question 7

What is proof strategy?

Accepted Answer

To prove an identity, start on one side (usually the more complicated) and transform it into the other using the identities above. Useful tactics:

Question 8

What is simplify using Pythagoras?

Accepted Answer

Simplify $\frac{1 - \cos^2 	heta}{\sin 	heta}$.

Question 9

What is prove an identity?

Accepted Answer

Prove $\sec^2 	heta - 1 = 	an^2 	heta$.

Question 10

What is treating $\sin 2	heta$ as $2 \sin 	heta$?

Accepted Answer

$\sin 2	heta = 2 \sin 	heta \cos 	heta$. The factor $\cos 	heta$ is essential.

Question 11

What is dropping the sign in $\cos 2	heta$?

Accepted Answer

All three forms ($\cos^2 - \sin^2$, $2 \cos^2 - 1$, $1 - 2 \sin^2$) are equivalent but easy to mis-write.

Question 12

What is forgetting the domain restriction?

Accepted Answer

Identities like $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$ require $\cos 	heta 
eq 0$. If a problem includes $	heta = \frac{\pi}{2}$, be careful.

Question 13

What is manipulating both sides simultaneously?

Accepted Answer

When proving an identity, do not start from the conclusion and work both sides; that is not a valid proof. Pick one side and transform it.

Question 14

What is sign errors from the quadrant?

Accepted Answer

When given a value of $\sin 	heta$ and a quadrant, the sign of $\cos 	heta$ (or $	an 	heta$) depends on the quadrant: positive in Q1; sine positive, cosine negative in Q2; both negative in Q3; cosine positive, sine negative in Q4.