A verification begins with $\sec x\tan x$. Rewrite every function in terms of sine and cosine. What results?

The working side is $\cot x\csc x$. After rewriting every function in terms of sine and cosine, which expression appears?

A student rewrites $\frac{\sin x}{\tan x}$ entirely in sine and cosine. Which is the correct next expression?

In a verification, rewrite $\frac{\cos x}{\cot x}$ entirely in sine and cosine. What is the result?

The expression $\sec x-\csc x$ is being rewritten entirely in sine and cosine. Which step is correct?

A verification starts with ${\tan }^{2}x+{\sec }^{2}x$. Rewrite all trigonometric functions in sine and cosine. Which expression results?

The current side is $\frac{1+\tan x}{\sec x}$. Rewriting each trig function in sine and cosine produces which expression?

A student rewrites $\frac{\csc x}{\cot x}$ using only sine and cosine. Which expression is correct?

Rewrite the working side ${\sec }^{2}x-{\tan }^{2}x$ entirely in sine and cosine. Which expression results?

A verification begins with $\cot x+\tan x$. Which expression results when both functions are rewritten in sine and cosine?

Midway through a verification, a student replaces $\tan x$ with $\frac{\sin x}{\cos x}$. Which type of identity licenses this substitution?

A printed step changes $\cot x$ to $\frac{\cos x}{\sin x}$. Which type of identity justifies that step?

In a verification, $\csc x$ is replaced by $\frac{1}{\sin x}$. Which type of identity was used?

A student changes $\sec x$ to $\frac{1}{\cos x}$ while verifying an identity. Which type of identity permits the change?

A verification replaces ${\sin }^{2}x+{\cos }^{2}x$ with 1. Which type of identity licenses this step?

A printed proof changes $1+{\tan }^{2}x$ into ${\sec }^{2}x$. Which type of identity justifies the substitution?

A student replaces $1+{\cot }^{2}x$ with ${\csc }^{2}x$. Which type of identity licenses that replacement?

In a verification, ${\sin }^{2}x$ becomes $1-{\cos }^{2}x$. Which identity supports this rearrangement?

The working side is $\frac{1}{1-\sin x}+\frac{1}{1+\sin x}$. Combine the fractions into one fraction. Which result is correct?

Combine $\frac{\sin x}{1+\cos x}+\frac{\sin x}{1-\cos x}$ over a common denominator. What is the next step?

The current expression is $\frac{\cos x}{1-\sin x}-\frac{\cos x}{1+\sin x}$. Which single fraction results after combining?

A verification has reached $\frac{1}{\sin x}+\frac{1}{\cos x}$. What is the correct common-denominator form?

Combine $\frac{1}{1+\tan x}+\frac{1}{1-\tan x}$ into one fraction. Which expression is correct?

The working side reads $\frac{\cos x}{1+\sin x}+\frac{\cos x}{1-\sin x}$. Which one-fraction step is valid?

A student combines $\frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}$. Which expression is the correct next step?

Combine $\frac{\sec x}{\sec x+\tan x}-\frac{\sec x}{\sec x-\tan x}$ over one denominator. What results?

To verify an identity, multiply the numerator and denominator of $\frac{\sin x}{1-\cos x}$ by the denominator's conjugate. What appears before simplifying?

A verification multiplies above and below in $\frac{\cos x}{1-\sin x}$ by the conjugate of its denominator. Which unsimplified expression results?

The fraction $\frac{1}{\sec x+\tan x}$ is rationalized by multiplying above and below by the conjugate. What is the immediate result?

To rationalize $\frac{1}{\csc x-\cot x}$, a student uses the denominator's conjugate. Which expression comes first?

A student multiplies $\frac{\tan x}{\sec x-1}$ above and below by the conjugate of the denominator. What is the unsimplified result?

The denominator of $\frac{\sin x}{1+\cos x}$ is rationalized with its conjugate. Which expression is written before simplification?

The expression ${\sin }^3\theta -\sin \theta {\cos }^2\theta$ is to be factored completely. Which next step is correct?

Factor the current side ${\cos }^3\theta +\cos \theta {\sin }^2\theta$ completely. Which expression results?

A verification reaches ${\tan }^{2}x-{\sec }^{2}x$ and factors it as a difference of squares. What is the correct factorization?

Factor ${\sin }^{2}x+\sin x\cos x$ as the next move in a verification. Which form is correct?

The expression ${\csc }^{2}x-{\cot }^{2}x$ is factored before using an identity. Which factorization is valid?

A student factors ${\sec }^{2}x+\sec x\tan x$. Which completely factored form is correct?

While simplifying $\frac{{\cos }^{2}x-1}{\cos x-1}$, a student cancels $\cos x$ and writes $\frac{\cos x-1}{-1}$. What went wrong?

A student writes $\sec x+\tan x=\frac{1+\sin x}{\cos x}$, then changes it to $\frac{1}{\cos x+\sin x}$. What is the error?

A proof uses $1-{\sin }^{2}x={\cos }^{2}x$, but the student writes $1+{\sin }^{2}x={\cos }^{2}x$. Which statement identifies the faulty step?

A student changes $\frac{\sin x+\cos x}{\sin x}$ into $1+\cos x$. What is wrong with this step?

To simplify $\frac{1-\cos x}{\sin x}$, a student multiplies only the numerator by $1+\cos x$. What is the error?

A student combines $\frac{1}{\sin x}+\frac{1}{\cos x}$ as $\frac{2}{\sin x+\cos x}$. What is the mistake?

To verify $\frac{1-{\cos }^{2}x}{\sin x}=\sin x$ by working on the left side, which opening move is most direct?

A student verifies $\frac{{\sec }^{2}x-1}{\tan x}=\tan x$ from the left side. What opening move reaches the target most efficiently?

For $\frac{{\csc }^{2}x-1}{\cot x}=\cot x$, which opening step on the left side is best?

To verify $\frac{{\sin }^{2}x}{1-\cos x}=1+\cos x$, what opening move on the left side is most useful?

A student works on the left side of $\frac{\sec x-\cos x}{\tan x}=\sin x$. Which opening move is most direct?

To verify $\frac{\tan x}{\sec x+1}=\frac{1-\cos x}{\sin x}$ from the left side, which opening move is most effective?