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Trigonometric Identities

Two new Pythagorean identities follow from sin²θ + cos²θ = 1 by dividing through. They link sec to tan and cosec to cot, and are essential tools for proofs and equations.

28 min Video by Zeeshan Zamurred Trigonometric Functions

Edexcel A Level Maths: 6.4 Trigonometric IdentitiesWatch the full walkthrough before the notes below.

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What you'll be able to do

Derive 1 + tan²θ = sec²θ
Derive 1 + cot²θ = cosec²θ
Use them to prove identities
Use them to solve equations

The two identities

Dividing $sin^{2} θ + cos^{2} θ = 1$ by $cos^{2} θ$ gives $tan^{2} θ + 1 = sec^{2} θ$ . Dividing by $sin^{2} θ$ gives $1 + cot^{2} θ = cosec^{2} θ$ .

1 + tan^{2} θ = sec^{2} θ, 1 + cot^{2} θ = cosec^{2} θ

The two derived Pythagorean identities.

Solving equations

Use an identity to write the equation in one function. For example, replace $sec^{2} θ$ with $1 + tan^{2} θ$ to get a quadratic in $tan θ$ .

1Replace

sec^{2} θ = 1 + tan^{2} θ

1 + tan^{2} θ = 3 + tan θ \Rightarrow tan^{2} θ - tan θ - 2 = 0

(tan θ - 2) (tan θ + 1) = 0 \Rightarrow tan θ = 2 or - 1

Answer

tan θ = 2

tan θ = - 1

Tip — Spot which identity converts the equation into a single trig function, then it becomes a quadratic.

Formula recap

1 + tan^{2} θ = sec^{2} θ

Divide by cos².

1 + cot^{2} θ = cosec^{2} θ

Divide by sin².

Common mistakes to avoid

Writing sec²θ = 1 − tan²θ.

It is 1 + tan²θ = sec²θ (a plus).

Mixing up which identity uses cot.

1 + cot²θ = cosec²θ (the “co” functions go together).

Key takeaways

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
Both come from sin²+cos²=1 by dividing.
Use them to reduce an equation to one trig function.

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