Maths Extension 1 Year 11 - Module 3 - Lesson 4
Pythagorean Identities
Name
Date
Class
1. Lesson focus
Pythagorean trigonometric identities. The identity sin²x+cos²x=1 generates 1+tan²x=sec²x and 1+cot²x=csc²x when the divisions are defined.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Simplify (sec²x−1)/tan²x, where defined.
3. Show that (1−cosx)(1+cosx)=sin²x.
3. Challenge
Solve sin²x=1/4 for 0≤x<2π.
4. Multiple choice
Which expression equals 1+cot²x wherever cot x is defined?
A. sin²x
B. sec²x
C. tan²x
D. csc²x
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Starting from sin²x+cos²x=1, derive an identity involving sec x and tan x.
Divide by cos²x (cosx≠0): tan²x+1=sec²x.
2. Simplify (sec²x−1)/tan²x, where defined.
Since sec²x−1=tan²x, the quotient is 1.
3. Show that (1−cosx)(1+cosx)=sin²x.
Left side=1−cos²x; by sin²x+cos²x=1, this equals sin²x.
Challenge. Solve sin²x=1/4 for 0≤x<2π.
sinx=±1/2, so x=π/6,5π/6,7π/6,11π/6.
Multiple choice. Which expression equals 1+cot²x wherever cot x is defined?
Correct answer: D. Divide sin²x+cos²x=1 by sin²x: 1+cot²x=csc²x.
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