📘 Notes

Proving Trigonometric Identities (AS Level 9709)

Basic identities you can use:

\( \sin^2 x + \cos^2 x = 1 \)
\( \tan x = \dfrac{\sin x}{\cos x} \)

Tip: When proving, convert everything to sines and cosines first, then simplify.

Example 1 — Show that \(1 + \tan^2 x = \dfrac{1}{\cos^2 x}\)

Start with \( \sin^2 x + \cos^2 x = 1 \).

Divide both sides by \( \cos^2 x \):

\( \dfrac{\sin^2 x}{\cos^2 x} + 1 = \dfrac{1}{\cos^2 x} \).

Replace \( \dfrac{\sin^2 x}{\cos^2 x} \) with \( \tan^2 x \):

\( \tan^2 x + 1 = \dfrac{1}{\cos^2 x} \).

Hence \( 1 + \tan^2 x = \dfrac{1}{\cos^2 x} \).

Example 2 — Prove that \( \cos^2 x = \dfrac{1}{1+\tan^2 x} \)

From Example 1 we know \( 1 + \tan^2 x = \dfrac{1}{\cos^2 x} \).

Take the reciprocal of both sides:

\( \dfrac{1}{1+\tan^2 x} = \cos^2 x \).

Thus \( \cos^2 x = \dfrac{1}{1+\tan^2 x} \).

Example 3 — Verify that \( \dfrac{\tan x}{1} = \dfrac{\sin x}{\cos x} \)

This is just a direct substitution definition.

By definition \( \tan x = \dfrac{\sin x}{\cos x} \).

So \( \tan x = \dfrac{\sin x}{\cos x} \) holds true.

Example 4 — Prove \( 1 - \cos^2 x = \sin^2 x \)

Start with \( \sin^2 x + \cos^2 x = 1 \).

Subtract \( \cos^2 x \) from both sides:

\( \sin^2 x = 1 - \cos^2 x \).

Hence \( 1 - \cos^2 x = \sin^2 x \).

Example 5 — Show that \( \dfrac{\sin^2 x}{\cos^2 x} = \tan^2 x \)

By definition \( \tan x = \dfrac{\sin x}{\cos x} \).

Square both sides:

\( \tan^2 x = \dfrac{\sin^2 x}{\cos^2 x} \).

Hence \( \dfrac{\sin^2 x}{\cos^2 x} = \tan^2 x \).

Checklist for Proving Identities

Write everything in terms of \( \sin x \) and \( \cos x \).
Use \( \sin^2 x + \cos^2 x = 1 \) to replace one square with the other.
Use algebra: factorising, simplifying fractions, or taking common denominators.
Work on one side only until it becomes the other side.

Practice Questions

Prove that \( \tan^2 x = \dfrac{1 - \cos^2 x}{\cos^2 x} \).
Show that \( 1 - \sin^2 x = \cos^2 x \).
Verify \( \dfrac{\sin^2 x}{1 - \cos^2 x} = 1 \) (where defined).

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Trigonometry — Trigonometric identities 38 problems