Trigonometry - Complete Basic Trigonometry Proofs

You can derive the formula $1+\tan^2\theta=\sec^2\theta$ by dividing the formula $\sin^2\theta+\cos^2\theta=1$ by which of the following functions?

You can derive the formula $1+\cot^2\theta=\csc^2\theta$ by dividing the formula $\sin^2\theta+\cos^2\theta=1$ by which of the following functions?

One popular way to simplify trigonometric expressions is to put the entire expression in terms of only ________ and ________ functions.

Fill in the four blanks, naming the appropriate trigonometric functions, and fill in the two question marks to complete the proofs showing that $\tan\theta=\frac{\sin\theta}{\cos\theta}$ and $\cot\theta=\frac{\cos\theta}{\sin\theta}$ .

For any angle $\theta$ :

____ $(\theta)=\frac{y}{r}$ , ____ $\theta=\frac{x}{r}$ , ____ $\theta=\frac{y}{x}$ , and ____ $\theta=\frac{x}{y}$ where is any point on the terminal side of $\theta$ and is units from the origin.

Then, $\tan\theta=?=\frac{\frac{y}{r}}{\frac{x}{r}}=\frac{\sin\theta}{\cos\theta}$ and $\cot\theta=?=\frac{\frac{x}{r}}{\frac{y}{r}}=\frac{\cos\theta}{\sin\theta}$

Which of the following might be an effective first step in verifying the following identity? Assume that you are going to attempt to convert the left hand side into the right hand side.

$\sqrt{\frac{\sec x-\tan x}{\sec x+\tan x}}=\frac{1}{\sec x+\tan x}$

Which of the following might be an effective first step in verifying the following identity? Assume that you are going to attempt to convert the left hand side into the right hand side.

$\frac{\sin\theta}{\csc \theta}+\frac{\cos \theta}{\sec\theta}=1$

Which of the following might be an effective first step in verifying the following identity? Assume that you are going to attempt to convert the left hand side into the right hand side.

$\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}$

Which of the following might be an effective first step in verifying the following identity? Assume that you are going to attempt to convert the left hand side into the right hand side.

$\tan^4\theta+\tan^2\theta=\sec^4\theta-\sec^2\theta$

Using trigonometric identities determine whether the following is valid:

$\tan2A = \frac{2\sin \left(-A\right ) \cos \left(-A\right )}{\cos^{2}\left(-A\right ) + \sin^{2}\left(-A\right)}$

Complete Basic Trigonometry Proofs

Practice Questions