Use the identities tan equals sin over cos and sin squared plus cos squared equals one to simplify expressions and prove simple results

What this dot point is asking

SEAB wants you to know the two basic trigonometric identities and use them to simplify expressions, find one ratio from another, and prove simple results. An identity is true for every value of the angle, unlike an equation, which is only true for particular angles. The two identities you need are the quotient identity and the Pythagorean identity, and almost every simplification leans on one of them.

The answer

The quotient identity

Tangent is defined as sine divided by cosine:

This lets you replace a tangent by a sine-over-cosine fraction, or the other way around, whenever it helps tidy an expression.

The Pythagorean identity

For every angle:

(The notation $\sin^2\theta$ means $(\sin\theta)^2$.) This comes straight from the unit circle: the point $(\cos\theta, \sin\theta)$ lies on a circle of radius $1$, and Pythagoras gives $x^2 + y^2 = 1$. Two rearrangements are constantly useful:

Finding one ratio from another

If you know one ratio and the quadrant, the Pythagorean identity gives the others. Solve for the square, take the square root, and choose the sign from the quadrant. The quotient identity then gives tangent.

Proving an identity

To prove an identity, work on one side only (usually the more complicated one) and use the two identities, plus ordinary algebra, until it matches the other side. Do not move terms across the equals sign as if solving an equation; instead transform one side step by step. Writing $\tan\theta$ as $\dfrac{\sin\theta}{\cos\theta}$ early often unlocks the proof.

Examples in context

Example 1. Simplifying before solving. Many trigonometric equations look hard until an identity simplifies them. Replacing $1 - \cos^2\theta$ with $\sin^2\theta$, for instance, can turn a mixed equation into one involving a single function, which is then solvable. So identities are the preparation step for the equation-solving topic.

Example 2. Right-triangle shortcut. If a right triangle gives $\sin\theta = \tfrac{3}{5}$, the identity immediately yields $\cos\theta = \tfrac{4}{5}$ without redrawing the triangle, and then $\tan\theta = \tfrac{3}{4}$. The identities let you move between ratios quickly in geometry and physics problems.

Try this

Q1. Given $\cos\theta = 0.8$ and $\theta$ acute, find $\sin\theta$. [2 marks]

• Cue. $\sin^2\theta = 1 - 0.64 = 0.36$, so $\sin\theta = 0.6$.

Q2. Simplify $1 - \cos^2\theta$. [1 mark]

• Cue. By the Pythagorean identity, $1 - \cos^2\theta = \sin^2\theta$.

Q3. Prove that $\tan\theta\,\cos\theta = \sin\theta$. [2 marks]

• Cue. Left side $= \dfrac{\sin\theta}{\cos\theta}\times\cos\theta = \sin\theta$, the right side.