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CIE A-Level Maths Study Notes

1.5.4 Trigonometric Identities

Trigonometric identities are crucial in mathematics, offering insights into the relationships between angles and their trigonometric functions. These identities are indispensable for solving complex problems in trigonometry, calculus, and beyond.

Fundamental Trigonometric Identities

The fundamental trigonometric identities include tan(θ)=sin(θ)cos(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} and sin2(θ)+cos2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. These are derived from the unit circle and the definitions of sine and cosine functions.

1. Tangent Identity:

tanθsinθcosθ\tan \theta \equiv \frac{\sin \theta}{\cos \theta}

This identity expresses tangent in terms of sine and cosine.


Simplify tanθ \tan \theta using the tangent identity when sinθ=35\sin \theta = \frac{3}{5} and cosθ=45\cos \theta = \frac{4}{5}.


tanθ=sinθcosθ=3545=34\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}

2. Pythagorean Identity

sin2θ+cos2θ1\sin^2 \theta + \cos^2 \theta \equiv 1

This identity reveals a fundamental relation between sine and cosine.


Prove the identity cos2(x)sin2(x)cos(x)+1cos(x)2cos(x)\frac{\cos^2(x) - \sin^2(x)}{\cos(x)} + \frac{1}{\cos(x)} \equiv 2 \cos(x).


Using sin2(x)=1cos2(x):\sin^2(x) = 1 - \cos^2(x):

=cos2(x)(1cos2(x))cos(x)+1cos(x)= \frac{\cos^2(x) - (1 - \cos^2(x))}{\cos(x)} + \frac{1}{\cos(x)}=2cos2(x)1cos(x)+1cos(x)= \frac{2\cos^2(x) - 1}{\cos(x)} + \frac{1}{\cos(x)}=2cos2(x)cos(x)= \frac{2\cos^2(x)}{\cos(x)} =2cos(x)= 2\cos(x)cos2(x)sin2(x)cos(x)+1cos(x)2cos(x)\therefore \frac{\cos^2(x) - \sin^2(x)}{\cos(x)} + \frac{1}{\cos(x)} \equiv 2 \cos(x)
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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