Use identities in trigonometric proof

Use identities in trigonometric proof

Using known trigonometric identities to prove other identities transforms one side of an equation into the other through algebraic and identity-based manipulation.

An identity is an equation that is true for every value of the variable for which both sides are defined. Proving a trigonometric identity is not the same as solving an equation: you do not find a value for the variable. Instead, you show that one expression can be rewritten, step by step, into the other.

The standard strategy

  • Work on one side only. Usually start with the more complicated side and simplify it until it matches the other side. Never treat the identity as an equation and move terms across the equals sign.
  • Convert everything to sines and cosines. This is often the quickest route when tangents, cotangents, secants or cosecants appear.
  • Combine fractions. Put terms over a common denominator, then add or simplify the numerators.
  • Factor and cancel. Look for common factors, difference of two squares, or expressions that simplify to 11.
  • Use the Pythagorean identities. These are the workhorses of most proofs.

Identities you should have ready

Quotient identities

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Reciprocal identities

sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ,cot⁡θ=1tan⁡θ\sec\theta = \frac{1}{\cos\theta}, \qquad \csc\theta = \frac{1}{\sin\theta}, \qquad \cot\theta = \frac{1}{\tan\theta}

Pythagorean identities

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 11+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

These can be rearranged, for example sin⁡2θ=1−cos⁡2θ\sin^2\theta = 1 - \cos^2\theta, which is useful for eliminating a squared term.

Useful algebraic moves

  • Difference of two squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), applied to expressions such as 1−sin⁡2θ1 - \sin^2\theta.
  • Common denominators: writing 1sin⁡θ+1cos⁡θ\frac{1}{\sin\theta} + \frac{1}{\cos\theta} as a single fraction.
  • Multiplying by a conjugate: multiplying numerator and denominator by 1+sin⁡θ1 + \sin\theta or 1−cos⁡θ1 - \cos\theta to create a Pythagorean identity.
  • Splitting fractions: sin⁡θ+cos⁡θcos⁡θ=tan⁡θ+1\frac{\sin\theta + \cos\theta}{\cos\theta} = \tan\theta + 1.

Presenting a proof

Write the proof as a chain of equivalent expressions, each line justified by an identity or an algebraic step. Keep the left-hand side and right-hand side labels clear, and finish by stating that the two sides are now identical. A tidy, well-labelled layout makes the reasoning easy to follow and easy to check.