Trigonometric Identities Made Simple: Algebra Tools for Trig Simplification

Why algebra matters more than “more identities”

Most trig simplification succeeds (or fails) because of algebra, not because you know a large list of identities. In this chapter you’ll practice the algebra moves that show up constantly: factoring, finding common denominators, simplifying rational expressions, using conjugates, and spotting difference of squares. The goal is to make your work mechanical and repeatable.

A repeatable simplification checklist

Use this order to avoid getting stuck or making expressions messier:

• 1) Rewrite (if needed): convert complex fractions to division, split numerators, or rewrite subtraction as addition of negatives. (Don’t reach for special trig substitutions yet.)
• 2) Factor: factor numerators and denominators completely (including factoring out common trig factors like sin(x)cos(x)).
• 3) Combine: if you have a sum/difference of fractions, get a common denominator and combine.
• 4) Cancel: cancel only common factors (never cancel terms across addition/subtraction).
• 5) Substitute only if it clearly helps: after the algebra is clean, decide whether a standard trig relationship will simplify what remains.

Factoring patterns you will use constantly

Factoring out a common trig factor

When every term shares a trig factor, pull it out first. This often exposes a simpler bracket that cancels later.

sin(x)cos(x) + sin(x) = sin(x)[cos(x) + 1]

Another common situation is a shared product:

sin^2(x)cos(x) - sin(x)cos(x) = sin(x)cos(x)[sin(x) - 1]

Difference of squares (algebra first, trig later)

Difference of squares is purely algebraic:

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a^2 - b^2 = (a - b)(a + b)

In trig work, a and b are often trig expressions:

cos^2(x) - sin^2(x) = (cos(x) - sin(x))(cos(x) + sin(x))

This is useful because factors like cos(x) - sin(x) may cancel with something elsewhere.

Factoring quadratics in trig expressions

Sometimes you’ll see something like a quadratic in sin(x) or cos(x). Treat it like a normal quadratic.

sin^2(x) - 3sin(x) + 2 = (sin(x) - 1)(sin(x) - 2)

The key is to view sin(x) as the “variable” during factoring.

Common denominators with trig fractions

Combining 1/sin(x) and 1/cos(x)

These show up constantly. The common denominator is sin(x)cos(x).

1/sin(x) + 1/cos(x) = cos(x)/[sin(x)cos(x)] + sin(x)/[sin(x)cos(x)] = [sin(x) + cos(x)]/[sin(x)cos(x)]

With subtraction:

1/sin(x) - 1/cos(x) = [cos(x) - sin(x)]/[sin(x)cos(x)]

Notice how the numerator becomes sin(x) ± cos(x) or cos(x) ± sin(x), which often factors or cancels later.

Mini-drill: common denominator only

Simplify each by combining into a single fraction (do not try to “finish” beyond that):

• 1/sin(x) + 2/cos(x)
• 3/sin(x) - 1/cos(x)
• 1/[sin(x)cos(x)] + 1/sin(x)

Answer format target: (...)/[sin(x)cos(x)].

Rational expressions: factor, then cancel

Canceling common factors (not terms)

Cancellation is legal only across multiplication. If you have addition/subtraction, factor first.

Example (factor then cancel):

[sin(x)cos(x)]/[sin(x)(1 + cos(x))]

Step-by-step:

• Factor check: numerator is already a product; denominator is sin(x)(1 + cos(x)).
• Cancel the common factor sin(x):

[sin(x)cos(x)]/[sin(x)(1 + cos(x))] = cos(x)/(1 + cos(x))

Non-example (illegal cancel):

(sin(x) + 1)/sin(x)  ≠  1 + 1

You cannot cancel the sin(x) because it is not a factor of the entire numerator; it’s only part of a sum.

Mini-drill: factor then cancel

Simplify by factoring out the greatest common factor first:

• [sin(x)cos(x) + sin(x)]/sin(x)
• [cos^2(x) - sin(x)cos(x)]/cos(x)
• [sin^2(x) + sin(x)cos(x)]/[sin(x)]

Complex fractions: clear the “fraction inside a fraction”

A complex fraction is something like (A/B)/(C/D) or a numerator/denominator that contains fractions. The most reliable method is to multiply top and bottom by the least common denominator (LCD) of all small denominators.

Example: dividing two trig fractions

(1/sin(x)) / (1/cos(x))

Rewrite division as multiplication by the reciprocal:

(1/sin(x)) / (1/cos(x)) = (1/sin(x)) * (cos(x)/1) = cos(x)/sin(x)

At this stage you’ve simplified the complex fraction using algebra alone.

Example: complex fraction with addition

[ (1/sin(x)) + (1/cos(x)) ] / [ (1/sin(x)) - (1/cos(x)) ]

Step-by-step using the checklist:

• Rewrite/Combine inside: combine each bracket using common denominators.

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

• Rewrite the big fraction:

[(sin + cos)/(sin cos)] / [(cos - sin)/(sin cos)]

• Factor/Cancel: multiply by the reciprocal and cancel the common factor sin(x)cos(x).

[(sin + cos)/(sin cos)] * [(sin cos)/(cos - sin)] = (sin + cos)/(cos - sin)

Notice how the trig product cancels cleanly because it was a factor, not a term.

Mini-drill: clear complex fractions

Simplify each by clearing denominators (aim to end with no “fractions inside fractions”):

• ( (2/sin(x)) / (3/cos(x)) )
• [ (1/cos(x)) ] / [ (sin(x)/cos(x)) ]
• [ 1 + (1/sin(x)) ] / [ 1 - (1/sin(x)) ]

Conjugates: when radicals or “1 ± something” appear

Conjugates are an algebra tool to simplify expressions involving a binomial with a plus/minus, especially when it creates a difference of squares.

Core idea

(a + b)(a - b) = a^2 - b^2

If you multiply a fraction by (a - b)/(a - b) (or (a + b)/(a + b)), you change its form without changing its value.

Example: removing a trig expression from a denominator

1 / (1 - cos(x))

Multiply by the conjugate:

1/(1 - cos) * (1 + cos)/(1 + cos) = (1 + cos)/[(1 - cos)(1 + cos)] = (1 + cos)/(1 - cos^2)

At this point you’ve used only algebra (conjugate + difference of squares). The denominator is now a single expression that is often easier to work with after your algebra steps are done.

Example: conjugate with a trig numerator

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

Multiply by the conjugate:

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

Again, the key move is that (1 + cos)(1 - cos) becomes a difference of squares.

Mini-drill: conjugate only

Rationalize/simplify the denominator using a conjugate (stop once the denominator becomes a difference of squares):

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

Integrated identity-style simplifications (algebra-led)

These problems are designed so that the main work is algebraic: factor, combine, cancel, and only then decide whether any substitution is helpful.

Problem 1: factor out a common trig factor, then cancel

[sin(x)cos(x) + sin(x)] / [sin^2(x) + sin(x)cos(x)]

Step-by-step:

• Factor numerator: sin(x)cos(x) + sin(x) = sin(x)[cos(x) + 1]
• Factor denominator: sin^2(x) + sin(x)cos(x) = sin(x)[sin(x) + cos(x)]
• Cancel common factor: cancel sin(x).

= [sin(cos + 1)]/[sin(sin + cos)] = (cos(x) + 1)/(sin(x) + cos(x))

Problem 2: combine fractions, then factor a difference of squares

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

Step-by-step:

• Common denominator: (cos - sin)(cos + sin)
• Combine:

= (cos + sin)/[(cos - sin)(cos + sin)] + (cos - sin)/[(cos - sin)(cos + sin)]

= [(cos + sin) + (cos - sin)] / [(cos - sin)(cos + sin)] = [2cos(x)]/[cos^2(x) - sin^2(x)]

You used the conjugate-style product automatically via difference of squares in the denominator.

Problem 3: simplify a complex fraction by multiplying by the LCD

[ (1/sin(x)) - (1/cos(x)) ] / [ 1/(sin(x)cos(x)) ]

Step-by-step:

• Rewrite division: dividing by 1/(sin cos) is multiplying by sin cos.

= ( (1/sin) - (1/cos) ) * (sin cos)

• Distribute and cancel factors:

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

One-move mini-drills (skill isolation)

Drill A: factor only

Factor completely (do not cancel or combine):

• sin(x)cos(x) - sin(x)
• cos^2(x) - sin^2(x)
• sin^2(x)cos(x) + sin(x)cos(x)

Drill B: common denominator only

Combine into one fraction:

• 1/sin(x) + 1/cos(x)
• 2/sin(x) - 3/cos(x)
• 1/(cos(x) - sin(x)) - 1/(cos(x) + sin(x))

Drill C: cancel only (after factoring)

Simplify by factoring first, then cancel common factors:

• [sin(x)(1 + cos(x))]/[sin(x)cos(x)]
• [cos(x)(cos(x) - sin(x))]/[cos^2(x) - sin(x)cos(x)]
• [sin(x)cos(x)]/[sin(x)(1 - cos(x))]

Drill D: conjugate only

Multiply by a conjugate to create a difference of squares in the denominator:

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

Common pitfalls to avoid (algebra rules that matter)

• Cancel factors, not terms: you may cancel sin(x) in sin(x)cos(x)/sin(x), but not in (sin(x)+1)/sin(x).
• Factor before you combine: if you see shared structure, factoring early often prevents huge denominators later.
• Use the LCD strategically: for complex fractions, multiplying top and bottom by the LCD is usually cleaner than combining everything first.
• Don’t substitute too early: keep expressions in factored form first; substitutions are most helpful after cancellation opportunities are exposed.