Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Preserve domain restrictions while simplifying a trigonometric quotient.
- Justify the conclusion "The original expression equals sin x for x≠kπ, k∈Z" using the stated assumptions.
Before you start
Pythagorean identity and division.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Simplify (1−cos²x)/sin x and state its domain.
Why this math matters
Preserve domain restrictions while simplifying a trigonometric quotient. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Angles are real and measured in radians.
- Equality is asserted only on the original domain.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
State where a simplified identity is actually defined
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Simplify (1−cos²x)/sin x and state its domain.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
1−cos²x=sin²x
Replace the numerator with an equivalent expression.
Work through the mathematics
sin²x/sin x=sin x only when sin x≠0
Cancellation requires a nonzero factor.
Check the conclusion
The original expression equals sin x for x≠kπ, k∈Z
The simplified formula can be defined at extra points, but those are not inputs of the original quotient.
The result
The original expression equals sin x for x≠kπ, k∈Z
The simplified formula can be defined at extra points, but those are not inputs of the original quotient.
Common mistakes to catch
- Cancellation does not fill a hole automatically.
- An identity must state any denominator exclusions.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is the original quotient defined at x=0?
Show a hint
Inspect its denominator.
Reveal answer and explanation
No
It has 0/0 there despite the simplified sine having value zero.
Practice 2
What is its value at x=π/2?
Show a hint
Substitute a nonexcluded input.
Reveal answer and explanation
One
Numerator and denominator both equal one.
Take the idea with you
Explain the difference between equivalent formulas on a domain and an extension to a larger domain.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: List every solution of a periodic equation
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.