Trigonometric Identities Made Simple: Reciprocal and Quotient Identities

Reciprocal Identities: Turning “Upside Down”

Reciprocal identities define three trig functions as the reciprocals of the basic ones. They are not “new” functions; they are shortcuts for writing 1/(sin), 1/(cos), and 1/(tan).

Practical use: Reciprocal identities are often used to (1) rewrite everything in terms of sin and cos, or (2) flip a fraction to simplify multiplication/division.

Example: Quick reciprocal rewrite

Simplify: sin x · csc x

sin x · csc x = sin x · (1/sin x) = 1

Quotient Identities: Ratios That Create tan and cot

Quotient identities express tan and cot as ratios of sin and cos. These are the main bridge for converting between “tan/cot language” and “sin/cos language.”

Example: Converting tan to sin/cos

Simplify: (tan x)(cos x)

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(tan x)(cos x) = (sin x / cos x)(cos x) = sin x

The “Default Baseline” Strategy: Rewrite Into sin and cos

A reliable baseline simplification strategy is: rewrite everything in terms of sin and cos, then simplify algebraically (cancel factors, combine fractions, factor common terms). This works because many identities and cancellations become visible only when everything is in a common language.

Here is the conversion map you’ll use most:

• sec x → 1/cos x
• csc x → 1/sin x
• tan x → sin x/cos x
• cot x → cos x/sin x

Example: Expression with sec and tan

Simplify: sec x · tan x

sec x · tan x = (1/cos x)(sin x/cos x) = sin x / cos^2 x

Notice the result is in sin and cos. That’s often the simplest baseline form unless you are specifically asked to express the result using sec or tan.

A Structured Rewrite Routine (Use This Every Time)

When simplifying trig expressions using reciprocal and quotient identities, follow this routine.

Step 1: Identify which trig functions appear

Circle or list the functions: sec, csc, cot, tan, etc. Decide whether converting to sin/cos will likely create cancellations.

Step 2: Convert to sin/cos when helpful

Use reciprocal and quotient identities to rewrite. A strong default is to convert everything to sin and cos, especially when you see products, quotients, or complex fractions.

Step 3: Factor common terms

Treat sin x and cos x like algebraic variables. Factor them when they appear in multiple terms.

Step 4: Reduce (cancel factors, combine fractions)

Cancel common factors (not terms across addition) and simplify complex fractions by multiplying numerator and denominator by a common factor.

Worked Examples (Step-by-Step)

Example 1: Simplify a quotient with sec

Simplify: sec x / tan x

Step 1: Functions: sec, tan  Step 2: Convert to sin/cos: sec x / tan x = (1/cos x) / (sin x/cos x)  Step 3: Rewrite division as multiplication: (1/cos x) · (cos x/sin x)  Step 4: Cancel cos x: = 1/sin x = csc x

This is a good example where converting to sin/cos reveals a clean cancellation and even returns to a single trig function.

Example 2: Simplify an expression with cot and csc

Simplify: cot x · sin x · csc x

Step 1: Functions: cot, sin, csc  Step 2: Convert: cot x = cos x/sin x, csc x = 1/sin x  Expression = (cos x/sin x) · sin x · (1/sin x)  Step 3: Factor/cancel: (cos x/sin x) · sin x = cos x  Step 4: Remaining factor: cos x · (1/sin x) = cos x/sin x = cot x

Even though we converted to sin/cos, the simplest final form is cot x.

Example 3: Simplify a complex fraction by converting to sin/cos

Simplify: 1 / (1 + tan x) by rewriting in sin and cos.

Step 1: Function: tan  Step 2: Convert: 1 / (1 + sin x/cos x)  Step 3: Combine inside denominator: 1 + sin x/cos x = (cos x + sin x)/cos x  Step 4: Divide by a fraction: 1 / ((cos x + sin x)/cos x) = cos x/(cos x + sin x)

Here, converting to sin/cos turns a “1 + tan” denominator into a single rational expression.

When NOT to Convert Everything: Efficiency Choices

Although “rewrite into sin/cos” is a great baseline, sometimes leaving tan or sec is shorter and cleaner. The key is to look for immediate cancellations or direct reciprocals.

Case A: Leaving tan is more efficient

Simplify: tan x · cot x

tan x · cot x = tan x · (1/tan x) = 1

If you converted both to sin/cos, you’d still get 1, but with extra steps.

Case B: Leaving sec is more efficient

Simplify: cos x · sec x

cos x · sec x = cos x · (1/cos x) = 1

Converting sec to 1/cos is already minimal; converting everything further doesn’t help.

Case C: Converting to sin/cos is clearly better

Simplify: tan x / sec x

tan x / sec x = (sin x/cos x) / (1/cos x) = (sin x/cos x)(cos x/1) = sin x

Here, the cos x cancels immediately after conversion.

Practice: Decide Whether to Convert to sin/cos

For each expression, (1) choose an efficient strategy (convert to sin/cos or keep tan/sec), and (2) simplify.

Set 1: Quick cancellations (often keep reciprocals/quotients)

• sec x · cos x
• csc x · sin x
• tan x · cot x
• (sec x)/(1/cos x)
• (1 + tan x) - tan x

Set 2: Convert to sin/cos to reveal cancellations

• (tan x)(cos x)
• (sec x)(sin x)(cos x)
• sec x / tan x
• cot x / csc x
• (tan x)/(sec x)

Set 3: Mixed strategy (choose what’s shorter)

• (sin x)/(sec x)
• (cos x)/(csc x)
• (sec x · tan x)/(tan x)
• (cot x · csc x)/(csc x)
• (1 - cot x)/(1/cot x)

Answer Key (Concise Steps)

Set 1

• sec x · cos x = 1
• csc x · sin x = 1
• tan x · cot x = 1
• (sec x)/(1/cos x) = (1/cos x)/(1/cos x) = 1
• (1 + tan x) - tan x = 1

Set 2

• (tan x)(cos x) = (sin x/cos x)(cos x) = sin x
• (sec x)(sin x)(cos x) = (1/cos x)(sin x)(cos x) = sin x
• sec x / tan x = (1/cos x)/(sin x/cos x) = 1/sin x = csc x
• cot x / csc x = (cos x/sin x)/(1/sin x) = cos x
• (tan x)/(sec x) = (sin x/cos x)/(1/cos x) = sin x

Set 3

• (sin x)/(sec x) = sin x · cos x (keeping sec as reciprocal is efficient)
• (cos x)/(csc x) = cos x · sin x
• (sec x · tan x)/(tan x) = sec x (cancel tan x)
• (cot x · csc x)/(csc x) = cot x (cancel csc x)
• (1 - cot x)/(1/cot x) = (1 - cot x)·cot x = cot x - cot^2 x