How to Write a Trigonometric Expression in Sine and Cosine

No specific trigonometric expression accompanies this instruction, so a single final answer cannot be determined. If you provide the expression, it can be simplified directly. The method below explains how to rewrite expressions using only sine and cosine and how to avoid changing where the original expression is defined.

The basic conversions

Use the quotient and reciprocal identities:

Function	Rewrite using sine and cosine
Tangent	$\tan x = \frac{\sin x}{\cos x}$
Cotangent	$\cot x = \frac{\cos x}{\sin x}$
Secant	$\sec x = \frac{1}{\cos x}$
Cosecant	$\csc x = \frac{1}{\sin x}$

After replacing these functions, simplify the resulting fractions. The Pythagorean identities are often useful:

$$\sin^2 x + \cos^2 x = 1, \qquad 1 - \sin^2 x = \cos^2 x, \qquad 1 - \cos^2 x = \sin^2 x.$$

The last two follow from the first. Choose the form that matches the factors in your expression; for example, $1-\cos^2 x$ can be replaced by $\sin^2 x$.

A reliable simplification method

1. Replace every tangent, cotangent, secant, and cosecant using the table.
2. Rewrite sums and differences over a common denominator if needed.
3. Factor the numerator and denominator before canceling anything.
4. Use a Pythagorean identity where it reduces the expression.
5. Check that you have not canceled a quantity that could be zero in the original expression.

It is usually safer to simplify one part at a time. If the expression contains nested fractions, rewrite the division explicitly and multiply by the reciprocal of the divisor. Do not cancel terms across addition or subtraction: cancellation applies to common factors, not to pieces of a sum.

Worked example

Rewrite and simplify

$$\frac{\sec x - \cos x}{\tan x}.$$

First replace secant and tangent with sine and cosine:

$$\begin{aligned} \frac{\sec x - \cos x}{\tan x} &= \frac{\frac{1}{\cos x} - \cos x}{\frac{\sin x}{\cos x}} \\ &= \frac{\frac{1-\cos^2 x}{\cos x}}{\frac{\sin x}{\cos x}}. \end{aligned}$$

Use $1-\cos^2 x=\sin^2 x$, then divide by a fraction by multiplying by its reciprocal:

$$\begin{aligned} \frac{\frac{\sin^2 x}{\cos x}}{\frac{\sin x}{\cos x}} &= \frac{\sin^2 x}{\cos x}\cdot\frac{\cos x}{\sin x} \\ &= \sin x. \end{aligned}$$

Thus the expression simplifies to $\sin x$ wherever the original expression is defined. In the original expression, $\sec x$ and $\tan x$ require $\cos x\ne 0$, and the denominator $\tan x$ must also be nonzero, so $\sin x\ne 0$. The simplified expression itself exists at more values of $x$; the equality is for the original expression’s domain.

Common mistakes

• Using the wrong quotient: $\tan x=\sin x/\cos x$, while $\cot x=\cos x/\sin x$.
• Canceling across a sum: In $(a+b)/a$, you cannot cancel the $a$ terms. Factor first, if possible.
• Dropping domain restrictions: A simplified expression can be defined at points where the original was not. Keep the original restrictions when describing equivalence.
• Changing the angle: Apply identities to matching angles. For instance, $1-\cos^2 x=\sin^2 x$ uses the same angle $x$ throughout.

If the task only asks for an expression in sine and cosine, it may not require reducing everything to a single function. For example, $\tan x+\sec x$ can be written as $\frac{\sin x}{\cos x}+\frac{1}{\cos x}$, then combined as $\frac{1+\sin x}{\cos x}$. Whether to simplify further depends on the exact expression and the requested form.