Other Pythagorean Identities
Divide the identity by sin² or cos².
Key idea
Divide the identity by sin² or cos².
In depth
Divide sin²θ + cos²θ = 1 by cos²θ to get tan²θ + 1 = sec²θ. Divide by sin²θ to get 1 + cot²θ = csc²θ. They are the same fact in different forms.
Worked example
If tan θ = 2, sec²θ = 5, so sec θ = ±√5.
Formula
Real-world uses
- Converting between tan and sec in calculus integrals and in engineering formulas
Common mistake
Writing 1 − tan² in place of 1 + tan².
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Other Pythagorean Identities?
Divide the identity by sin² or cos². It is a topic at the advanced level in Trigonometry.
What is the formula for Other Pythagorean Identities?
The key formula is 1 + tan²θ = sec²θ.
Where is Other Pythagorean Identities used in real life?
Real-world examples: Converting between tan and sec in calculus integrals and in engineering formulas.
What should I learn before Other Pythagorean Identities?
Make sure you are comfortable with Pythagorean Identity, the previous topic in Trigonometry.
What is a common mistake with Other Pythagorean Identities?
Writing 1 − tan² in place of 1 + tan².
How can I practise Other Pythagorean Identities?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Other Pythagorean Identities?
Move on to Quotient Identities next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.