Trigonometric Identities Made Simple: Core Simplification Patterns You Will See Often

This chapter is a recognition toolbox: you will see a “shape,” make a reliable first move, and simplify with minimal wandering. Each pattern below includes (a) when it appears, (b) the best first move, (c) a worked example, and (d) a quick back-check using a known identity so you can verify your result immediately.

Tool 1: Mixed tan and sec expressions → rewrite in sin/cos

(a) When it appears

Any time you see sums/differences/products involving tan x and sec x together, especially with fractions like tan x / sec x, sec x - tan x, or rational expressions in sec and tan. These often collapse quickly after converting to sin/cos.

(b) Best first move

Convert everything to sin x and cos x immediately: tan x = sin x / cos x and sec x = 1 / cos x. Then simplify like an algebraic fraction.

(c) Worked example (before → after)

Before: (tan x + sec x)/(sec x - tan x)

First move: rewrite in sin/cos.

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

Combine terms over the common denominator cos x in numerator and denominator:

= ((sin x + 1)/cos x)/((1 - sin x)/cos x)

Cancel the common factor 1/cos x:

= (sin x + 1)/(1 - sin x)

After: (1 + sin x)/(1 - sin x)

(d) Practice + identity-based back-check

Practice: Simplify (sec x + tan x)/sec x.

Do it:

(sec x + tan x)/sec x = sec x/sec x + tan x/sec x = 1 + tan x/sec x

Now convert the remaining ratio using tan x/sec x:

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

Answer: 1 + sin x

Back-check (identity-based): Replace tan x and sec x by sin/cos in the original: ((1/cos)+(sin/cos))/(1/cos) = ((1+sin)/cos)/(1/cos)=1+sin. Matches.

Tool 2: Conjugate products with sec and tan

(a) When it appears

Products of the form (sec x - tan x)(sec x + tan x), or any (A-B)(A+B) where A and B are trig expressions. This is a “difference of squares” trigger.

(b) Best first move

Use the algebra pattern (A-B)(A+B)=A^2-B^2, then apply the Pythagorean relationship connecting sec and tan (already known from earlier work): sec^2 x - tan^2 x = 1.

(c) Worked example (before → after)

Before: (sec x - tan x)(sec x + tan x)

First move: difference of squares.

(sec x - tan x)(sec x + tan x) = sec^2 x - tan^2 x

Now use the identity:

sec^2 x - tan^2 x = 1

After: 1

(d) Practice + identity-based back-check

Practice: Simplify (sec x + tan x)(sec x - tan x) + (tan^2 x - sec^2 x).

Do it: The first product is sec^2 x - tan^2 x. So the whole expression becomes

(sec^2 x - tan^2 x) + (tan^2 x - sec^2 x) = 0

Back-check (identity-based): Since sec^2 x - tan^2 x = 1, the expression is 1 + (-1) = 0. Verified.

Tool 3: The “(1 - cos x)/sin x” and “sin x/(1 + cos x)” family (half-angle look without using half-angles)

(a) When it appears

Any time you see (1 - cos x) or (1 + cos x) sitting over sin x, or sin x over (1 ± cos x). These are classic rationalization targets and often simplify to a single trig ratio.

(b) Best first move

Multiply numerator and denominator by the conjugate of the 1 ± cos x expression to create 1 - cos^2 x, which becomes sin^2 x. This is valid because you are multiplying by (1 cos x)/(1 cos x), i.e., by 1 (where defined).

(c) Worked example (before → after)

Before: (1 - cos x)/sin x

First move: multiply by the conjugate (1 + cos x)/(1 + cos x).

(1 - cos x)/sin x * (1 + cos x)/(1 + cos x) = (1 - cos^2 x)/(sin x(1 + cos x))

Use 1 - cos^2 x = sin^2 x:

= sin^2 x/(sin x(1 + cos x))

Cancel a factor of sin x:

= sin x/(1 + cos x)

After: sin x/(1 + cos x)

Why this is a simplification: you traded a “difference over sine” for a “sine over sum,” which often cancels nicely in larger expressions.

(d) Practice + identity-based back-check

Practice: Simplify sin x/(1 - cos x).

Do it (conjugate):

sin x/(1 - cos x) * (1 + cos x)/(1 + cos x) = sin x(1 + cos x)/(1 - cos^2 x)

Replace 1 - cos^2 x with sin^2 x and cancel:

= sin x(1 + cos x)/sin^2 x = (1 + cos x)/sin x

Answer: (1 + cos x)/sin x

Back-check (identity-based): Multiply your result by (1 - cos x)/(1 - cos x) to see if you return to the original:

((1 + cos x)/sin x) * ((1 - cos x)/(1 - cos x)) = (1 - cos^2 x)/(sin x(1 - cos x)) = sin^2 x/(sin x(1 - cos x)) = sin x/(1 - cos x)

Matches.

Tool 4: Spot-and-cancel with 1 sin x or 1 cos x after rationalizing

(a) When it appears

Rational expressions like 1/(1 - sin x), 1/(1 + sin x), 1/(1 - cos x). These often look “stuck” until you create a difference of squares in the denominator.

(b) Best first move

Multiply by the conjugate to turn the denominator into 1 - sin^2 x or 1 - cos^2 x, then use 1 - sin^2 x = cos^2 x or 1 - cos^2 x = sin^2 x.

(c) Worked example (before → after)

Before: 1/(1 - sin x)

First move: multiply by (1 + sin x)/(1 + sin x).

1/(1 - sin x) * (1 + sin x)/(1 + sin x) = (1 + sin x)/(1 - sin^2 x)

Use 1 - sin^2 x = cos^2 x:

= (1 + sin x)/cos^2 x

After: (1 + sin x)/cos^2 x (often written as (1 + sin x)sec^2 x if that helps later).

(d) Practice + identity-based back-check

Practice: Simplify 1/(1 + cos x).

Do it:

1/(1 + cos x) * (1 - cos x)/(1 - cos x) = (1 - cos x)/(1 - cos^2 x) = (1 - cos x)/sin^2 x

Answer: (1 - cos x)/sin^2 x

Back-check (identity-based): Multiply your simplified form by (1 + cos x)/(1 + cos x):

((1 - cos x)/sin^2 x) * ((1 + cos x)/(1 + cos x)) = (1 - cos^2 x)/(sin^2 x(1 + cos x)) = sin^2 x/(sin^2 x(1 + cos x)) = 1/(1 + cos x)

Matches.

Tool 5: “Divide by cos” or “divide by sin” to trigger a known identity

(a) When it appears

Expressions like sin^2 x + cos^2 x sitting inside a fraction, especially (sin^2 x + cos^2 x)/cos^2 x or (sin^2 x + cos^2 x)/sin^2 x. Also sums like 1 + tan^2 x or 1 + cot^2 x may be hiding in a fraction.

(b) Best first move

If everything is over cos^2 x, split the fraction and rewrite as tan^2 x + 1. If everything is over sin^2 x, rewrite as 1 + cot^2 x. This is valid because you are dividing each term by the same nonzero quantity and using the quotient definitions.

(c) Worked example (before → after)

Before: (sin^2 x + cos^2 x)/cos^2 x

First move: split and simplify term-by-term.

(sin^2 x + cos^2 x)/cos^2 x = sin^2 x/cos^2 x + cos^2 x/cos^2 x = tan^2 x + 1

Now use the identity 1 + tan^2 x = sec^2 x:

= sec^2 x

After: sec^2 x

(d) Practice + identity-based back-check

Practice: Simplify (sin^2 x + cos^2 x)/sin^2 x.

Do it:

(sin^2 x + cos^2 x)/sin^2 x = 1 + cos^2 x/sin^2 x = 1 + cot^2 x

Use 1 + cot^2 x = csc^2 x:

= csc^2 x

Back-check (identity-based): Replace csc^2 x by 1/sin^2 x and compare: since sin^2 x + cos^2 x = 1, the original equals 1/sin^2 x, which is csc^2 x.

Tool 6: Nested fractions with tan/sec  clear denominators, then convert

(a) When it appears

Complex fractions like 1/(sec x + tan x), (1 - sin x)/(cos x) mixed with sec, or any “fraction inside a fraction” involving sec and tan. These are often one conjugate away from collapsing.

(b) Best first move

Use a conjugate if you see sec x tan x in a denominator; otherwise clear denominators by multiplying numerator and denominator by the common denominator. Then (if needed) convert to sin/cos.

(c) Worked example (before → after)

Before: 1/(sec x + tan x)

First move: multiply by the conjugate (sec x - tan x)/(sec x - tan x).

1/(sec x + tan x) * (sec x - tan x)/(sec x - tan x) = (sec x - tan x)/(sec^2 x - tan^2 x)

Use sec^2 x - tan^2 x = 1:

= sec x - tan x

After: sec x - tan x

(d) Practice + identity-based back-check

Practice: Simplify 1/(sec x - tan x).

Do it:

1/(sec x - tan x) * (sec x + tan x)/(sec x + tan x) = (sec x + tan x)/(sec^2 x - tan^2 x) = sec x + tan x

Back-check (identity-based): Multiply your simplified answer by the original denominator: (sec x + tan x)(sec x - tan x) = sec^2 x - tan^2 x = 1, so indeed sec x + tan x is the reciprocal of sec x - tan x.

Quick pattern index (recognize  first move  typical result)