B. Trigonometric Functions · Trigonometric Identities and Equations

Proving Identities: Reciprocal, Quotient and Pythagorean

B3.3About 70 minutes

Learning goals

I can write a correct two-column (L.S./R.S.) proof without moving terms across the equals sign.
I can choose a strategy: start with the more complicated side, rewrite in sine and cosine, combine fractions, factor.
I can use the Pythagorean identity in all its forms, including 1 + tan²x = sec²x and 1 + cot²x = csc²x.
I can multiply by a conjugate such as (1 + cos x)/(1 + cos x) when a proof needs it.

Words to know

proof
A chain of correct steps, each justified by algebra or a known identity, that shows one side of an equation equals the other.

L.S. and R.S.
Short for left side and right side. In a proof you simplify each side separately until both are the same expression.

conjugate
For a binomial a + b, the conjugate is a − b. Their product a² − b² often turns into a Pythagorean identity, e.g. (1 − cos x)(1 + cos x) = sin²x.

difference of squares
The factoring pattern a² − b² = (a − b)(a + b).

Why proofs?

In the last lesson you saw that testing values can only suggest an identity. A proof shows it is true for every x in the domain. Proofs also train the skill that matters most in university mathematics: building a careful argument where every step is justified.

Remember from Grade 11: you proved identities in degrees in the unit Trigonometric Ratios, Identities and Laws. The rules are the same in radians. This lesson sharpens the strategies; the next adds the compound and double angle formulas.

The point (cos x, sin x) on the unit circle makes a right triangle with legs cos x and sin x and hypotenuse 1. Dividing sin²x + cos²x = 1 by cos²x or by sin²x gives the other two Pythagorean identities.

The rules of a proof

An identity is what you are trying to prove, so you cannot treat it like an equation to solve. In particular, you may not add, subtract, multiply or divide both sides by something, because that assumes the two sides are already equal.

Instead, write two columns, L.S. and R.S. Simplify one side (or each side separately) until both columns show the same expression. Then write L.S. = R.S. You may:

substitute any known identity;
do algebra on one side: expand, factor, find a common denominator;
multiply one side by a clever form of 1, such as (1 + cos x)/(1 + cos x).

A strategy for proving identities. Most proofs use steps 1 to 4 in this order; a conjugate helps when a denominator is 1 ± sin x or 1 ± cos x.

Worked example

Example 1: Rewrite in sine and cosine

Prove csc x − sin x = cos x cot x.

The left side has a reciprocal, so start there. L.S. = 1/sin x − sin x.
Common denominator: = (1 − sin²x)/sin x.
Pythagorean identity: = cos²x/sin x.
Split the product: = cos x · (cos x/sin x) = cos x cot x = R.S.

L.S. = R.S., so the identity is proved (for x ≠ kπ).

Worked example

Example 2: Pythagorean identities in a fraction

Prove (1 + tan²x)/(1 + cot²x) = tan²x.

L.S. = sec²x/csc²x, using both Pythagorean identities.
= (1/cos²x) ÷ (1/sin²x) = (1/cos²x) × sin²x.
= sin²x/cos²x = tan²x = R.S.

L.S. = R.S. ✓ (for x ≠ kπ/2).

Worked example

Example 3: Using a conjugate

Prove sin x/(1 − cos x) = (1 + cos x)/sin x.

The left denominator 1 − cos x has the conjugate 1 + cos x. Multiply the L.S. by (1 + cos x)/(1 + cos x), which equals 1.
L.S. = sin x(1 + cos x)/((1 − cos x)(1 + cos x)) = sin x(1 + cos x)/(1 − cos²x).
= sin x(1 + cos x)/sin²x.
Cancel one sin x: = (1 + cos x)/sin x = R.S.

L.S. = R.S. ✓ (for x ≠ kπ).

Worked example

Example 4: Working on both sides

Prove tan x + cot x = sec x csc x.

L.S. = sin x/cos x + cos x/sin x = (sin²x + cos²x)/(sin x cos x) = 1/(sin x cos x).
R.S. = (1/cos x)(1/sin x) = 1/(sin x cos x).
Both sides simplify to the same expression, and we never moved anything across the equals sign.

L.S. = R.S. ✓ (for x ≠ kπ/2).

Worked example

Example 5: Factoring

Prove tan²x − sin²x = tan²x sin²x.

L.S. = sin²x/cos²x − sin²x.
Common factor sin²x: = sin²x(1/cos²x − 1) = sin²x · (1 − cos²x)/cos²x.
= sin²x · sin²x/cos²x = (sin²x/cos²x) · sin²x = tan²x sin²x = R.S.

L.S. = R.S. ✓ (for x ≠ π/2 + kπ).

Before you start

Test the identity at one value (say x = 1 in radian mode) before spending time on a proof. If the sides differ, it is not an identity, and no proof exists. If they agree, look at both sides and ask: which side has more going on (fractions, reciprocals, squares)? Start there.

Just for fun

Working on both sides of an identity until they meet is like digging a tunnel from both ends. The two crews digging the Channel Tunnel between England and France met under the sea in 1990. In a proof, just make sure your two ends meet at exactly the same expression.

Hands-on activity

Proof puzzle

You need: a sheet of paper, scissors (use them carefully), a pencil and a partner (a parent, sibling or friend).

Step 1. Copy each line of the proof of Example 3 onto its own strip of paper, without numbering them:

L.S. = sin x/(1 − cos x)
= sin x(1 + cos x)/((1 − cos x)(1 + cos x))
= sin x(1 + cos x)/(1 − cos²x)
= sin x(1 + cos x)/sin²x
= (1 + cos x)/sin x = R.S.

Step 2. Shuffle the strips. Your partner puts them in order, and you explain the reason for each step (multiply by the conjugate, difference of squares, Pythagorean identity, cancel).

Step 3. Swap roles with Example 5.

What to look for: the person explaining should name a reason for every step without looking at the lesson. If a step cannot be justified, that is exactly the step to practise.

Your turn

Prove sin x sec x = tan x.

Prove cos x(sec x − cos x) = sin²x.

Prove (1 − cos²x)(1 + cot²x) = 1.

Prove sec x − sin x tan x = cos x.

Prove 1/(1 + sin x) + 1/(1 − sin x) = 2 sec²x.

Prove sin⁴x − cos⁴x = 1 − 2cos²x.

More practice: warm-up

Write sec x in terms of cos x.

Write cot x in terms of sin x and cos x.

Simplify tan x csc x.

Simplify 1 − sin²x.

Simplify sec²x − 1.

More practice: on your own

Prove (sin x + cos x)² + (sin x − cos x)² = 2.

Prove cos x/(1 + sin x) = sec x − tan x.

Prove csc²x + sec²x = csc²x sec²x.

Prove (1 + cot x)/csc x = sin x + cos x.

Prove (tan x − sin x)/sin³x = sec x/(1 + cos x).

Challenge

Prove (1 − cos x)/(1 + cos x) = (csc x − cot x)².

Prove sec⁴x − tan⁴x = sec²x + tan²x.

Prove sin x/(1 − cot x) + cos x/(1 − tan x) = sin x + cos x.

Check your understanding

Quiz

What is usually the best first step when proving an identity?

Which step is NOT allowed in a proof of an identity?

Which expression is equal to sec x − cos x?

What should you multiply sin x/(1 − sin x) by to use a conjugate?

Which identity comes from dividing sin²x + cos²x = 1 by sin²x?

Simplify (1 + tan²x) cos²x.

For parents and teachers

Big idea (B3.3): to prove an identity, simplify one side (or each side separately) until both are the same; never move terms across the equals sign. The main tools are the reciprocal, quotient and Pythagorean identities, common denominators, factoring and conjugates.

How to help: Ask your teen to read a proof aloud and give the reason for every line ("Pythagorean identity", "common denominator"). If they are stuck, ask: "Which side looks messier? Can you write it all in sine and cosine?"

Common difficulties:

Treating the identity like an equation (cross-multiplying, adding to both sides).
Fraction errors when finding a common denominator.
Forgetting the other forms of the Pythagorean identity (1 − cos²x = sin²x, sec²x − 1 = tan²x).
Giving up too early: proofs often look worse before they look better.