Trigonometric Identities Made Simple: Double-Angle and Half-Angle Connections

Building the Double-Angle Identities from Sum Identities

The double-angle identities come directly from the angle-sum formulas by using the same angle twice. This is powerful because it turns expressions involving 2x into expressions involving x, which are often easier to simplify or evaluate.

Deriving sin(2x)

Start from the sum identity for sine and set both angles equal to x:

sin(a + b) = sin a cos b + cos a sin b

Let a = x and b = x:

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

Result:

sin(2x) = 2 sin x cos x

Deriving cos(2x)

Start from the sum identity for cosine and again set both angles equal to x:

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cos(a + b) = cos a cos b − sin a sin b

Let a = x and b = x:

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

Base double-angle form:

cos(2x) = cos^2 x − sin^2 x

Why cos(2x) Has Multiple Equivalent Forms

The identity cos(2x) = cos^2 x − sin^2 x is only the starting point. By rewriting either sin^2 x or cos^2 x using a Pythagorean relationship, you can create versions that are better suited to a particular expression.

All common equivalent forms

How the alternate forms are obtained (step-by-step)

Starting from:

cos(2x) = cos^2 x − sin^2 x

To get cos(2x) = 1 − 2sin^2 x: replace cos^2 x with 1 − sin^2 x.

cos(2x) = (1 − sin^2 x) − sin^2 x = 1 − 2sin^2 x

To get cos(2x) = 2cos^2 x − 1: replace sin^2 x with 1 − cos^2 x.

cos(2x) = cos^2 x − (1 − cos^2 x) = 2cos^2 x − 1

Choosing the Best cos(2x) Form: A Practical Decision Rule

• If the expression contains sin^2 x and you want to remove it, use sin^2 x = (1 − cos 2x)/2 or rearrange from cos(2x) = 1 − 2sin^2 x.
• If the expression contains cos^2 x and you want to remove it, use cos^2 x = (1 + cos 2x)/2 or rearrange from cos(2x) = 2cos^2 x − 1.
• If you see both squares and no clear target, cos^2 x − sin^2 x may simplify immediately (for example, when paired with sin^2 x + cos^2 x elsewhere).

Example 1: Eliminate sin^2 x

Simplify: 3 + 2cos^2 x − 2sin^2 x.

Step 1 (spot the pattern): cos^2 x − sin^2 x appears.

Step 2 (use the difference-of-squares form):

3 + 2(cos^2 x − sin^2 x) = 3 + 2cos(2x)

Result: 3 + 2cos(2x).

Example 2: Convert 1 − 2sin^2 x into cos(2x)

Simplify: 5(1 − 2sin^2 x) − 1.

Step 1 (recognize the exact form): 1 − 2sin^2 x = cos(2x).

Step 2 (substitute):

5cos(2x) − 1

Example 3: Convert 2cos^2 x − 1 into cos(2x)

Simplify: 4cos^2 x − 2.

Step 1 (factor):

4cos^2 x − 2 = 2(2cos^2 x − 1)

Step 2 (use the identity): 2cos^2 x − 1 = cos(2x).

2cos(2x)

Connecting to Half-Angle Forms by Solving for Squares

Half-angle identities are best understood as a “reverse move” from the cosine double-angle formulas. Instead of starting with x and producing 2x, we start with a statement about cos(2u) and solve for sin^2 u or cos^2 u. Then we set u = x/2.

Derive sin^2(x/2)

Start with:

cos(2u) = 1 − 2sin^2 u

Step 1 (isolate sin^2 u):

2sin^2 u = 1 − cos(2u)

sin^2 u = (1 − cos(2u))/2

Step 2 (substitute u = x/2, so 2u = x):

sin^2(x/2) = (1 − cos x)/2

Derive cos^2(x/2)

Start with:

cos(2u) = 2cos^2 u − 1

Step 1 (isolate cos^2 u):

2cos^2 u = 1 + cos(2u)

cos^2 u = (1 + cos(2u))/2

Step 2 (substitute u = x/2):

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

Sign Considerations: Getting sin(x/2) and cos(x/2) (Not Squared)

The half-angle results above give sin^2(x/2) and cos^2(x/2). If you take square roots to get sin(x/2) or cos(x/2), you must choose the correct sign based on the quadrant of x/2 (not the quadrant of x).

sin(x/2) = ± sqrt((1 − cos x)/2)

cos(x/2) = ± sqrt((1 + cos x)/2)

How to choose the sign (quadrant context)

• Determine where the angle x/2 lies (Quadrant I, II, III, or IV), using the given information about x.
• Use the sign of sine/cosine in that quadrant to pick + or −.

Example: Choosing the correct sign

Suppose x = 300°. Then x/2 = 150°, which is in Quadrant II. In Quadrant II, sine is positive and cosine is negative.

Compute sin(150°) using the half-angle formula:

sin(150°) = sin(300°/2) = + sqrt((1 − cos 300°)/2)

Since cos 300° = 1/2:

sin(150°) = + sqrt((1 − 1/2)/2) = sqrt((1/2)/2) = sqrt(1/4) = 1/2

Compute cos(150°):

cos(150°) = − sqrt((1 + cos 300°)/2) = − sqrt((1 + 1/2)/2) = − sqrt((3/2)/2) = − sqrt(3/4) = −(sqrt 3)/2

Exercises: Choose the Best Double-Angle Form

In each exercise, your goal is not just to simplify, but to choose the most efficient form of cos(2x) to eliminate either sin^2 x or cos^2 x (or to recognize cos^2 x − sin^2 x).

A. Identify the best form (no full simplification required)

• 1) To eliminate sin^2 x from 7 − 4sin^2 x, which form is best: cos^2 x − sin^2 x, 1 − 2sin^2 x, or 2cos^2 x − 1?
• 2) To eliminate cos^2 x from 3 + 6cos^2 x, which form is best?
• 3) To simplify 5(cos^2 x − sin^2 x), which form is best?
• 4) To rewrite sin^2 x in terms of cos(2x) inside sin^2 x + 2, which rearranged identity is best?

B. Simplify (show the key substitution step)

• 5) Simplify: 9 − 6sin^2 x.
• 6) Simplify: 8cos^2 x − 4.
• 7) Simplify: 2 + (cos^2 x − sin^2 x).
• 8) Simplify: 1 − 2sin^2 x + 2cos^2 x − 1.

C. Mixed practice: choose a form to eliminate a square

• 9) Simplify: sin^2 x + sin^2 x cos(2x) by rewriting cos(2x) in a form involving sin^2 x.
• 10) Simplify: cos^2 x(1 − cos(2x)) by rewriting cos(2x) in a form involving cos^2 x.

D. Half-angle square practice (no sign ambiguity)

• 11) Rewrite in terms of cos x: sin^2(x/2) + cos^2(x/2).
• 12) Rewrite in terms of cos x: 3sin^2(x/2) − 2cos^2(x/2).

E. Half-angle with sign (quadrant context)

• 13) If x = 240°, find sin(x/2) and cos(x/2) using half-angle formulas and quadrant signs.
• 14) If x is in Quadrant III, determine the sign of cos(x/2) and sin(x/2) (state which quadrants x/2 could lie in).