Section 1
Basic Identities
If we know the value of $\sin \theta,$ is it possible to find the other five trigonometric function values? If not, what other information is needed?
a. Explain how the identities $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $\cot ^{2} \theta+1=\csc ^{2} \theta$ can be derived from the identity $\cos ^{2} \theta+\sin ^{2} \theta=1$b. The identity $\cos ^{2} \theta+\sin ^{2} \theta=1$ is true for all real numbers. Are the identities $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $\cot ^{2} \theta+1=\csc ^{2} \theta$ also true for all real numbers? Explain your answer.
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\tan \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\cot \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\sec \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\csc \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\cot \theta \sec \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\tan ^{2} \theta+1$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\cot ^{2} \theta+1$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\tan \theta \sec \theta \cot \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\frac{1}{\sec \theta \csc \theta}$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\frac{\tan \theta}{\cot \theta}+\tan \theta \cot \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\frac{1}{\tan \theta}+\cot \theta$$
In $3-14,$ write each expression as a single term using $\sin \theta, \cos \theta,$ or both.$$\sec \theta+\frac{1}{\csc \theta}$$