A trigonometric identity is an equation that involves trigonometric functions and holds true for all valid values of the angle. In this topic, we derive three fundamental Pythagoras identities using right-angled triangles and unit circles: \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), and \(1 + \cot^2 \theta = \text{cosec}^2 \theta\). We then apply these identities to evaluate trigonometric expressions without using a calculator and to verify and prove various trigonometric identities.
Visualise how the three basic identities connect directly to Pythagoras' theorem \(a^2 + b^2 = c^2\).
Start with \(a^2 + b^2 = c^2\).
Divide by \(c^2\): \(\sin^2 A + \cos^2 A = 1\).
Divide by \(b^2\): \(1 + \tan^2 A = \sec^2 A\).
Divide by \(a^2\): \(1 + \cot^2 A = \text{cosec}^2 A\).
Identity 1
Divide \(a^2 + b^2 = c^2\) by \(c^2\).
Identity 2
Divide \(a^2 + b^2 = c^2\) by \(b^2\).
Identity 3
Divide \(a^2 + b^2 = c^2\) by \(a^2\).
Unit Circle Form
Where \(x = \cos \theta\) and \(y = \sin \theta\).
All basic trigonometric identities stem directly from Pythagoras' theorem.
A trigonometric identity is an equation involving trigonometric functions that is valid for any values of the angle. Reciprocal and quotient identities: \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\), \(\cot \theta = \dfrac{1}{\tan \theta} = \dfrac{\cos \theta}{\sin \theta}\), \(\text{cosec } \theta = \dfrac{1}{\sin \theta}\), \(\sec \theta = \dfrac{1}{\cos \theta}\).
From a right-angled triangle \(ABC\) with sides \(a\), \(b\), and hypotenuse \(c\) where angle \(A\) is considered: \(\sin A = \dfrac{a}{c}\), \(\cos A = \dfrac{b}{c}\), \(\tan A = \dfrac{a}{b}\), \(\text{cosec } A = \dfrac{c}{a}\), \(\sec A = \dfrac{c}{b}\), \(\cot A = \dfrac{b}{a}\). By Pythagoras' theorem: \(a^2 + b^2 = c^2\).
Using rearrangement of basic identities: \(\sin^2 (-430^\circ) + \cos^2 (-430^\circ) = 1\) since \(\sin^2 \theta + \cos^2 \theta = 1\) for any angle \(\theta\). Also, \(\tan^2 \left(\dfrac{\pi}{3}\right) - \sec^2 \left(\dfrac{\pi}{3}\right) = -1\) since \(1 + \tan^2 \theta = \sec^2 \theta \implies \tan^2 \theta - \sec^2 \theta = -1\).
Proofs can be done by simplifying the expression on the left-hand side (LHS) until it matches the right-hand side (RHS), simplifying the RHS until it matches the LHS, or simplifying both LHS and RHS independently until both arrive at the exact same simplified expression.
Prove from the more complex side.
Convert expressions to basic trigonometric ratios form (in terms of \(\sin\) and \(\cos\)).
Multiply by a conjugate if required, e.g. when terms like \(1 + \cos \theta\) or \(1 - \sin \theta\) appear in denominators.
Combine fractions over a common denominator and substitute basic identities directly.
Example 18(a): Prove \(1 - 2\sin^2 A = 2\cos^2 A - 1\)
Substitute \(\sin^2 A = 1 - \cos^2 A\).
Example 18(b): Prove \(\tan A + \cot A = \sec A \text{ cosec } A\)
Combine fractions using \(\sin^2 A + \cos^2 A = 1\).
Answer in your mind, then press "Check Answer".
Without using a calculator, find the value of \(\sec^2 173^\circ - \tan^2 173^\circ\).
Given \(\cos \theta = m\), express \(\tan^2 \theta\) in terms of \(m\).
Prove the identity: \(\dfrac{\sin^2 \theta}{1 + \cos \theta} = 1 - \cos \theta\).
Trigonometric identities are only valid for acute angles between \(0^\circ\) and \(90^\circ\).
A trigonometric identity is valid for any angle \(\theta\) for which the functions are defined, including obtuse, reflex, and negative angles (e.g., \(\sin^2(-430^\circ) + \cos^2(-430^\circ) = 1\)).
You must always work exclusively from the left-hand side to prove an identity.
You can prove from LHS to RHS, from RHS to LHS, or simplify both LHS and RHS independently to the same final expression.
Objective: Test your understanding of basic trigonometric identities and their applications.
1 Which of the following is equivalent to \(\text{cosec}^2 \theta - \cot^2 \theta\)?
2 What is the result when both sides of Pythagoras' theorem \(a^2 + b^2 = c^2\) are divided by \(b^2\)?
3 Based on the basic trigonometric identities, the numerical value of \(\tan^2 \left(\dfrac{\pi}{3}\right) - \sec^2 \left(\dfrac{\pi}{3}\right)\) is ________.
4 Dividing both sides of the equation \(a^2 + b^2 = c^2\) by \(a^2\) derives the identity \(1 + \cot^2 A =\) ________.
Match each trigonometric expression to its simplified equivalent or derived identity. On a phone, tap a term first, then tap the matching box.
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