Simple Trigonometric Identities

The Quotient Identity states that the tangent of an angle is equal to the sine of that angle divided by its cosine: $\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$

This identity is derived from the right-triangle definitions where $\sin \theta = \frac{O}{H}$ and $\cos \theta = \frac{A}{H}$. Dividing these two ratios results in $\frac{O}{A}$, which is the definition of $\tan \theta$.

It is particularly useful when an equation contains both $\tan \theta$ and $\sin \theta$ or $\cos \theta$, as it allows the entire equation to be expressed in terms of just sine and cosine.

The Pythagorean Identity is expressed as: $\sin^2 \theta + \cos^2 \theta \equiv 1$

This identity is a direct consequence of Pythagoras' Theorem ($a^2 + b^2 = c^2$) applied to a unit circle or a right triangle. In a right triangle with hypotenuse $1$, the legs are $\sin \theta$ and $\cos \theta$, leading to the sum of their squares equaling $1^2$.

It is frequently rearranged to substitute squared terms: $\sin^2 \theta \equiv 1 - \cos^2 \theta$ or $\cos^2 \theta \equiv 1 - \sin^2 \theta$. These forms are essential for solving quadratic trigonometric equations.

Substitution: This is the primary method for simplifying expressions. If you see $\tan \theta$, consider replacing it with $\frac{\sin \theta}{\cos \theta}$. If you see $\sin^2 \theta$, consider replacing it with $1 - \cos^2 \theta$ to create a uniform equation.

Elimination of Fractions: When using the quotient identity in an equation, multiplying through by $\cos \theta$ is a common step to remove denominators and linearize the expression.

Quadratic Transformation: In equations involving both $\cos x$ and $\sin^2 x$, use the Pythagorean identity to convert the $\sin^2 x$ term into terms of $\cos x$. This results in a quadratic equation in the form $a\cos^2 x + b\cos x + c = 0$, which can then be factored or solved using the quadratic formula.

Check the Formula Booklet: Simple identities like $\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$ and $\sin^2 \theta + \cos^2 \theta \equiv 1$ are often NOT provided in exam formula booklets; they must be memorized perfectly.

Look for 'Show That' Clues: If a question asks you to 'show that' one expression equals another, look at the target expression. If the target has no tangent terms, you must substitute $\tan \theta$ out early in your working.

Common Factorization: After substituting identities, look for common factors. For example, in $2\sin \theta \cos \theta = \sin \theta$, do not divide by $\sin \theta$ (which loses solutions); instead, rearrange to $2\sin \theta \cos \theta - \sin \theta = 0$ and factorize.

Verify the Range: After solving an equation using identities, always ensure your final angles fall within the specific interval requested (e.g., $0 \le \theta < 360$ or $-\pi \le \theta < \pi$).