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Edexcel A Level Maths: 6.4 Trigonometric Identities Curated by Kepler, not produced by us. Watch it before the notes below.
Derive 1 + tan²θ = sec²θ Derive 1 + cot²θ = cosec²θ Use them to prove identities Use them to solve equations 1
The two identities Dividing sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 cos 2 θ \cos^2\theta cos 2 θ tan 2 θ + 1 = sec 2 θ \tan^2\theta + 1 = \sec^2\theta tan 2 θ + 1 = sec 2 θ sin 2 θ \sin^2\theta sin 2 θ 1 + cot 2 θ = cosec 2 θ 1 + \cot^2\theta = \operatorname{cosec}^2\theta 1 + cot 2 θ = cosec 2 θ
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Solving equations Use an identity to write the equation in one function. For example, replace sec 2 θ \sec^2\theta sec 2 θ 1 + tan 2 θ 1 + \tan^2\theta 1 + tan 2 θ tan θ \tan\theta tan θ
Example 1 Solve sec 2 θ = 3 + tan θ \sec^2\theta = 3 + \tan\theta sec 2 θ = 3 + tan θ 0 ≤ θ < 2 π 0 \le \theta < 2\pi 0 ≤ θ < 2 π 1 Replace sec 2 θ = 1 + tan 2 θ \sec^2\theta = 1 + \tan^2\theta sec 2 θ = 1 + tan 2 θ 2 1 + tan 2 θ = 3 + tan θ ⇒ tan 2 θ − tan θ − 2 = 0 1 + \tan^2\theta = 3 + \tan\theta \Rightarrow \tan^2\theta - \tan\theta - 2 = 0 1 + tan 2 θ = 3 + tan θ ⇒ tan 2 θ − tan θ − 2 = 0 3 ( tan θ − 2 ) ( tan θ + 1 ) = 0 ⇒ tan θ = 2 or − 1 (\tan\theta - 2)(\tan\theta + 1) = 0 \Rightarrow \tan\theta = 2 \text{ or } -1 ( tan θ − 2 ) ( tan θ + 1 ) = 0 ⇒ tan θ = 2 or − 1 Answer tan θ = 2 \tan\theta = 2 tan θ = 2 or
tan θ = − 1 \tan\theta = -1 tan θ = − 1 Tip — Spot which identity converts the equation into a single trig function, then it becomes a quadratic.
1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ Divide by cos². 1 + cot 2 θ = cosec 2 θ 1 + \cot^2\theta = \operatorname{cosec}^2\theta 1 + cot 2 θ = cosec 2 θ Divide by sin².
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).
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.
Answer a few questions, then review anything you missed.
No lives limit · learn from each explanation Three lives · three mistakes end the run Prefer paper? Open the written self-check Answer each question before opening its answer. This self-check does not record a quiz score.
The identity for sec is:
1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ 1 − tan 2 θ = sec 2 θ 1 - \tan^2\theta = \sec^2\theta 1 − tan 2 θ = sec 2 θ 1 + cot 2 θ = sec 2 θ 1 + \cot^2\theta = \sec^2\theta 1 + cot 2 θ = sec 2 θ tan 2 θ = sec 2 θ \tan^2\theta = \sec^2\theta tan 2 θ = sec 2 θ Show answer and explanation Answer: 1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ
Divide by cos².
The identity for cosec is:
1 + cot 2 θ = cosec 2 θ 1 + \cot^2\theta = \operatorname{cosec}^2\theta 1 + cot 2 θ = cosec 2 θ 1 + tan 2 θ = cosec 2 θ 1 + \tan^2\theta = \operatorname{cosec}^2\theta 1 + tan 2 θ = cosec 2 θ 1 − cot 2 θ = cosec 2 θ 1 - \cot^2\theta = \operatorname{cosec}^2\theta 1 − cot 2 θ = cosec 2 θ cot 2 θ = cosec 2 θ \cot^2\theta = \operatorname{cosec}^2\theta cot 2 θ = cosec 2 θ Show answer and explanation Answer: 1 + cot 2 θ = cosec 2 θ 1 + \cot^2\theta = \operatorname{cosec}^2\theta 1 + cot 2 θ = cosec 2 θ
Divide by sin².
These identities are derived from:
sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 the sine rule small angle approx the cosine rule Show answer and explanation Answer: sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1
By dividing.
sec 2 θ − tan 2 θ \sec^2\theta - \tan^2\theta sec 2 θ − tan 2 θ
1 1 1 0 0 0 2 2 2 tan 2 θ \tan^2\theta tan 2 θ Show answer and explanation Answer: 1 1 1
Rearrange the identity.
Replacing sec 2 θ \sec^2\theta sec 2 θ sec 2 θ = 3 + tan θ \sec^2\theta = 3 + \tan\theta sec 2 θ = 3 + tan θ
tan 2 θ − tan θ − 2 = 0 \tan^2\theta - \tan\theta - 2 = 0 tan 2 θ − tan θ − 2 = 0 tan 2 θ + tan θ − 2 = 0 \tan^2\theta + \tan\theta - 2 = 0 tan 2 θ + tan θ − 2 = 0 tan 2 θ − tan θ + 2 = 0 \tan^2\theta - \tan\theta + 2 = 0 tan 2 θ − tan θ + 2 = 0 tan 2 θ = 3 \tan^2\theta = 3 tan 2 θ = 3 Show answer and explanation Answer: tan 2 θ − tan θ − 2 = 0 \tan^2\theta - \tan\theta - 2 = 0 tan 2 θ − tan θ − 2 = 0
1 + tan² = 3 + tan.
Keep practising Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.
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