Trig Identities

The 0606 identity kit (all memorised, none given):

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
• $\sin^2\theta + \cos^2\theta = 1$, and $\div\cos^2\theta$: $1 + \tan^2\theta = \sec^2\theta$; $\div\sin^2\theta$: $1 + \cot^2\theta = \csc^2\theta$
• Reciprocals: $\sec\theta = \dfrac{1}{\cos\theta}$, $\csc\theta = \dfrac{1}{\sin\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$ ($= \dfrac{\cos\theta}{\sin\theta}$)

Learn the divided forms as divisions of the core identity, one memory generating three identities is a smaller attack surface for exam-day blanks.

Proofs: the one-side discipline

“Prove that $(1 - \cos^2\theta)(1 + \cot^2\theta) = 1$”, the prove command demands a one-directional transformation:

LHS $= \sin^2\theta \times \csc^2\theta$ (Pythagorean rearrangement; reciprocal form) $= \sin^2\theta \times \dfrac{1}{\sin^2\theta} = 1 =$ RHS $\blacksquare$

Rules of the genre: start from one side (usually the messier), transform line by line, never move terms across the identity as if solving an equation, and land exactly on the other side. Each manipulation on its own line, the per-step marks need per-step visibility.

The two universal moves

Convert everything to sin and cos. When no route is visible, rewrite $\tan$, $\cot$, $\sec$, $\csc$ in $\sin/\cos$ terms and simplify the fractions. This single move completes most 0606 proofs, it’s the default gear, not the last resort.

Spot Pythagorean fragments. The examiners’ favourite disguise: $1 - \sin^2\theta$ is $\cos^2\theta$; $\sec^2\theta - 1$ is $\tan^2\theta$; $\sin^2\theta - 1$ is $-\cos^2\theta$. Any ”$1 \pm$ squared-trig” or “squared-trig $\pm$ squared-trig” expression should trigger the check. Fraction proofs often add one more step: combine over a common denominator, then the fragment appears.

Identities as equation-solvers

Beyond proofs, the identities convert mixed equations into single-function ones: $2\cos^2 x + \sin x = 1$ $\to$ substitute $\cos^2 x = 1 - \sin^2 x$ $\to$ a quadratic in $\sin x$. The substitution line is the method mark. The same kit simplifies expressions before differentiation when a trig-calculus question looks worse than it is.

Common mistakes

• Working both sides simultaneously, or cross-multiplying the identity
• The divided Pythagorean forms misremembered (sign or pairing errors)
• Fragments unspotted: $1 - \cos^2\theta$ expanded into nothing useful
• Three manipulations crushed into one line, unawardable if any slip occurs
• $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$ forgotten as the often-shorter form

Full topic context: Trigonometry notes · the exam-technique drill: trig technique guide.