Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. They are tools for rewriting, simplifying, and transforming expressions and equations in trigonometry, algebra, and later calculus.

In this chapter, the emphasis is on recognizing and using the main types of trigonometric identities, not on graphing or on the basic definitions of the trigonometric functions themselves.

What a Trigonometric Identity Is (and Is Not)

An identity is an equation that is always true (on its domain), such as
$$
\sin^2\theta + \cos^2\theta = 1.
$$
It holds for every angle $\theta$ for which $\sin\theta$ and $\cos\theta$ are defined.

In contrast, a trigonometric equation like
$$
\sin\theta = \tfrac{1}{2}
$$
is not an identity; it is true only for specific angles (for example, $\theta = 30^\circ, 150^\circ$, plus other coterminal angles).

When working with trigonometric identities, you never “solve for $\theta$.” Instead, you:

• Show that two expressions are equivalent.
• Rewrite one expression into a simpler or more useful form.

Basic Trigonometric Identities

The main identities can be grouped into several families. Later sections in this part of the course will focus on specific important ones (like Pythagorean and sum–difference identities), but here we list the overall landscape.

Reciprocal Identities

These express some trig functions as reciprocals of others:
$$
\csc\theta = \frac{1}{\sin\theta}, \quad
\sec\theta = \frac{1}{\cos\theta}, \quad
\cot\theta = \frac{1}{\tan\theta}.
$$

Equivalently,
$$
\sin\theta = \frac{1}{\csc\theta}, \quad
\cos\theta = \frac{1}{\sec\theta}, \quad
\tan\theta = \frac{1}{\cot\theta}.
$$

They tell you that once you know $\sin\theta$ and $\cos\theta$ (and where they are nonzero), you can get all the other basic trigonometric functions.

Quotient Identities

These show $\tan$ and $\cot$ in terms of $\sin$ and $\cos$:
$$
\tan\theta = \frac{\sin\theta}{\cos\theta} \quad (\cos\theta \neq 0),
$$
$$
\cot\theta = \frac{\cos\theta}{\sin\theta} \quad (\sin\theta \neq 0).
$$

These are especially useful for rewriting expressions so that they involve only $\sin$ and $\cos$.

Pythagorean Identities (Overview)

The fundamental Pythagorean identity is
$$
\sin^2\theta + \cos^2\theta = 1.
$$
From it, you can derive:
$$
1 + \tan^2\theta = \sec^2\theta, \quad
1 + \cot^2\theta = \csc^2\theta.
$$

These will be studied more closely in the “Pythagorean identities” subsection, but for this chapter it is important to recognize them as central tools for simplifying expressions involving squared trig functions.

Even–Odd Identities

These describe how trig functions behave with negative angles:

• Sine, tangent, and cotangent are odd functions:
  $$
  \sin(-\theta) = -\sin\theta, \quad
  \tan(-\theta) = -\tan\theta, \quad
  \cot(-\theta) = -\cot\theta.
  $$
• Cosine, secant, cosecant are even or odd as follows:
  $$
  \cos(-\theta) = \cos\theta, \quad
  \sec(-\theta) = \sec\theta, \quad
  \csc(-\theta) = -\csc\theta.
  $$

You can use these to remove negatives from inside trig functions or to compare values at opposite angles.

Co-function Identities (Degrees Form)

Co-function identities relate functions of complementary angles (angles adding to $90^\circ$). In degrees:
$$
\sin(90^\circ - \theta) = \cos\theta, \quad
\cos(90^\circ - \theta) = \sin\theta,
$$
$$
\tan(90^\circ - \theta) = \cot\theta, \quad
\cot(90^\circ - \theta) = \tan\theta,
$$
$$
\sec(90^\circ - \theta) = \csc\theta, \quad
\csc(90^\circ - \theta) = \sec\theta.
$$

In radians, $90^\circ$ is $\tfrac{\pi}{2}$, so for example
$$
\sin\left(\frac{\pi}{2} - \theta \right) = \cos\theta.
$$

Co-function identities are particularly helpful when evaluating trig expressions at complementary angles or rewriting them in terms of a single trig function.

Periodicity Identities (Overview)

Trigonometric functions repeat their values at regular intervals called periods. In radians:

• Sine and cosine have period $2\pi$:
  $$
  \sin(\theta + 2\pi) = \sin\theta, \quad
  \cos(\theta + 2\pi) = \cos\theta.
  $$
• Tangent and cotangent have period $\pi$:
  $$
  \tan(\theta + \pi) = \tan\theta, \quad
  \cot(\theta + \pi) = \cot\theta.
  $$

These identities allow you to replace an angle by a coterminal angle to simplify evaluation.

Structure of More Advanced Identities

Beyond these basic forms, there are several important families of identities that relate trig functions of angle combinations. In this chapter, the goal is to see how they fit conceptually, not to derive them in full detail.

Sum and Difference Identities (Conceptual Overview)

These express trig functions of sums or differences of angles (like $\alpha + \beta$) in terms of trig functions of the individual angles $\alpha$ and $\beta$.

For sine:
$$
\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta.
$$

For cosine:
$$
\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta.
$$

For tangent, the pattern involves fractions. These will be treated in detail in their own subsection, but you should recognize that:

• They let you compute values like $\sin 75^\circ$ or $\cos 15^\circ$ exactly by breaking them into sums/differences of “friendly” angles.
• They also underlie many other identities like double-angle and half-angle identities.

Double-Angle Identities

These are special cases of sum identities in which the two angles are equal ($\alpha = \beta$), giving expressions for functions of $2\theta$ in terms of $\theta$.

For sine:
$$
\sin(2\theta) = 2\sin\theta \cos\theta.
$$

For cosine, there are several equivalent forms:
$$
\cos(2\theta) = \cos^2\theta - \sin^2\theta
= 2\cos^2\theta - 1
= 1 - 2\sin^2\theta.
$$

For tangent:
$$
\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
\quad \text{(when the denominator is nonzero)}.
$$

Double-angle identities are used to:

• Rewrite expressions involving $2\theta$ in terms of $\theta$.
• Express powers like $\sin^2\theta$ and $\cos^2\theta$ in alternative forms.

Half-Angle Identities

Half-angle identities go in the opposite direction: they express $\sin(\tfrac{\theta}{2})$ or $\cos(\tfrac{\theta}{2})$ in terms of $\cos\theta$ (and sometimes $\sin\theta$). Typical forms use square roots, such as
$$
\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}, \quad
\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{2}.
$$

After taking square roots, you choose a plus or minus sign depending on the quadrant of $\tfrac{\theta}{2}$.

Half-angle identities are frequently used to simplify integrals in calculus and to find exact trig values for some angles.

Product-to-Sum and Sum-to-Product Identities

These identities convert products of trig functions into sums or differences, and vice versa. Schematically:

• Product-to-sum:
• $\sin\alpha \sin\beta$, $\cos\alpha \cos\beta$, and $\sin\alpha \cos\beta$ can be rewritten as combinations of $\cos(\alpha \pm \beta)$ and $\sin(\alpha \pm \beta)$.
• Sum-to-product:
• Expressions like $\sin\alpha + \sin\beta$ or $\cos\alpha - \cos\beta$ can be written as products involving sines and cosines of average and half-difference of the angles.

These identities help simplify complicated trigonometric expressions, especially when dealing with sums of oscillatory terms (for example, in physics or signal processing).

General Strategies for Using Trigonometric Identities

When you are asked to “verify” or “simplify using identities,” the goal is to transform an expression using known identities until it matches a given target or becomes simpler.

Some common strategies:

1. Work on one side only.
   For identity verification, you usually take the more complicated side and manipulate it until it matches the other side. Do not treat the identity as an equation to solve by doing the same operation to both sides.
2. Rewrite everything in terms of $\sin$ and $\cos$ when in doubt.
   Use the reciprocal and quotient identities to replace $\tan$, $\cot$, $\sec$, $\csc$ with $\sin$ and $\cos$. This often reveals cancellations or makes Pythagorean identities easier to apply.
3. Look for squares or combinations suggesting Pythagorean identities.
   When you see $\sin^2\theta + \cos^2\theta$, or something that can be rearranged into that, replace it with $. Similarly, look for patterns that match + \tan^2\theta$ or + \cot^2\theta$.
4. Use even–odd and periodic identities to simplify angles.
   For example, replace $\sin(-\theta)$ with $-\sin\theta$, or reduce angles by adding/subtracting full periods (like \pi$ or $\pi$) to get standard-equivalent angles.
5. Factor and simplify algebraically.
   Many identity problems rely on basic algebra: factoring, expanding, simplifying fractions, and canceling common factors. Trig identities are often mixed with ordinary algebra.
6. Use angle-sum and double-angle identities when angles are sums or multiples.
   When you see expressions like $\sin(2\theta)$ or $\cos(\alpha + \beta)$, consider expanding them using the appropriate angle identities so you can combine like terms or match target forms.

Typical Types of Problems Involving Identities

You will meet several styles of exercises that rely on trigonometric identities:

• Verification:
  Show that one expression is identically equal to another by transforming one side into the other using known identities.
• Simplification:
  Rewrite an expression (such as a complicated fraction of trig functions) into a simpler or more compact form.
• Rewriting in terms of a single function:
  Express everything, for example, in terms of $\sin\theta$ only, or only $\cos\theta$, using reciprocal, quotient, and Pythagorean identities.
• Evaluation using identities:
  Compute exact values like $\sin 75^\circ$ or $\cos \tfrac{\pi}{12}$ by expressing them as sums/differences of standard angles and then using sum–difference identities.
• Solving trig equations (with identities):
  Use identities to transform an equation into a simpler one, then solve for angles. (The detailed solving process belongs to trigonometric equations, but identities are a key tool in that process.)

Understanding the main families of trigonometric identities and how they interrelate gives you a flexible toolbox for handling trigonometric expressions throughout later mathematics.