Pythagorean Identities
The three Pythagorean identities are among the most frequently used equations in trigonometry, and they all spring from the Pythagorean theorem applied to the unit circle.
The foundational identity is sin²θ + cos²θ = 1. Since any point on the unit circle has coordinates (cosθ, sinθ) and the radius is 1, this equation is simply x² + y² = 1 in disguise.
Dividing both sides by cos²θ produces the second identity: tan²θ + 1 = sec²θ. Dividing the original by sin²θ yields the third: 1 + cot²θ = csc²θ.
Memorize the first identity and you can derive the other two on the spot by dividing.
These identities are indispensable for simplifying expressions, solving equations, and proving other identities. For instance, if you encounter 1 − sin²θ in an expression, you can instantly replace it with cos²θ. Similarly, sec²θ − 1 simplifies to tan²θ.
In calculus, Pythagorean identities are essential for integration techniques like trigonometric substitution, where expressions like √(1 − x²) are rewritten using x = sinθ.
Reciprocal and Quotient Identities
The six trigonometric functions are really just three pairs of reciprocals, plus two quotient relationships.
The reciprocal identities are: cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ. These are straightforward but easy to mix up under exam pressure, so commit them to memory.
The quotient identities express tangent and cotangent in terms of sine and cosine: tanθ = sinθ/cosθ and cotθ = cosθ/sinθ.
Convert everything to sine and cosine
These identities serve as the workhorse tools for simplifying complex trig expressions. A common strategy when verifying an identity is to convert everything to sine and cosine using the reciprocal and quotient identities, then simplify.
To simplify secθ · sinθ, rewrite as (1/cosθ) · sinθ = sinθ/cosθ = tanθ.
tanθSimilarly, cotθ · secθ = (cosθ/sinθ) · (1/cosθ) = 1/sinθ = cscθ.
cscθMastering this conversion technique eliminates confusion when faced with expressions involving four or more different trig functions. Whenever you feel stuck on a trig simplification, this "convert to sin and cos" approach almost always opens a path forward.
Double Angle Formulas
The double angle formulas express trig functions of 2θ in terms of functions of θ.
For sine: sin(2θ) = 2 sinθ cosθ.
For cosine, there are three equivalent forms: cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
For tangent: tan(2θ) = 2tanθ / (1 − tan²θ).
The cosine double angle formula is especially versatile because its three forms let you choose whichever version contains the function you already know.
If you know sinθ = 3/5 and need cos(2θ), use 1 − 2sin²θ = 1 − 2(9/25) = 1 − 18/25 = 7/25 without ever finding cosθ.
cos(2θ) = 7/25Power-reducing formula in calculus
Double angle formulas appear constantly in calculus, particularly when integrating powers of sine and cosine. The identity cos²θ = (1 + cos2θ)/2 (a rearrangement of the double angle formula) is called the power-reducing formula and is essential for evaluating integrals like ∫cos²x dx.
If you need to verify a double angle calculation quickly, Solver AI can walk you through each substitution step by step.
Half Angle Formulas
The half angle formulas let you find the sine, cosine, or tangent of half an angle when you know the full angle. They are derived directly from the double angle formulas by solving for the single-angle function.
The formulas are: sin(θ/2) = ±√((1 − cosθ)/2) and cos(θ/2) = ±√((1 + cosθ)/2). The ± sign depends on the quadrant in which θ/2 falls.
For tangent: tan(θ/2) = sinθ/(1 + cosθ) = (1 − cosθ)/sinθ.
These alternative tangent forms avoid the square root and the ± ambiguity, making them preferable in most calculations.
To find sin(15°), recognize that 15° = 30°/2. Using the half angle formula: sin(15°) = √((1 − cos30°)/2) = √((1 − √3/2)/2) = √((2 − √3)/4). Since 15° is in the first quadrant, sine is positive.
sin(15°) = √((2 − √3)/4)Half angle formulas are particularly useful in integral calculus for the Weierstrass substitution t = tan(θ/2), which converts any rational expression in sine and cosine into a rational function of t that can be integrated using partial fractions. This technique handles integrals that no other method can easily solve.
Sum and Difference Formulas
The sum and difference formulas allow you to find exact values of trig functions for angles that are sums or differences of standard angles.
The formulas are: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB (note the sign flips for cosine), and tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).
These are extremely useful for finding exact values of non-standard angles.
cos(75°) = cos(45° + 30°) = cos45° cos30° − sin45° sin30° = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4.
cos(75°) = (√6 − √2)/4The sum formulas also serve as the foundation from which the double angle formulas are derived (set A = B = θ in the sum formula to get the double angle).
Applications in physics and engineering
In physics and engineering, these formulas are crucial for analyzing wave interference: when two waves sin(ωt) and sin(ωt + φ) combine, the sum formula determines whether they reinforce or cancel each other.
Recognizing when an expression matches the pattern of a sum or difference formula is a key skill that saves time on exams and helps you simplify otherwise intractable expressions.
How to Verify Trigonometric Identities
Verifying a trig identity means showing that the left side of an equation equals the right side for all valid values of the variable. The golden rule: work on one side only and transform it until it matches the other side.
Never cross-multiply or move terms across the equals sign — that assumes the equation is true, which is what you are trying to prove.
Here are proven strategies:
- Start with the more complex side
It is usually easier to simplify than to complicate.
- Convert everything to sine and cosine
Use reciprocal and quotient identities.
- Factor whenever possible
Look for common factors or Pythagorean identity patterns like
sin²θ + sin θ = sinθ(sinθ + 1). - Combine fractions
Place them over a common denominator.
- Multiply by a strategic form of 1
Such as
(1 + cosθ)/(1 + cosθ), to create a difference of squares.
To verify sinθ/(1 − cosθ) = cscθ + cotθ, multiply the left side's numerator and denominator by (1 + cosθ) to get sinθ(1 + cosθ)/sin²θ = (1 + cosθ)/sinθ = 1/sinθ + cosθ/sinθ = cscθ + cotθ.
cscθ + cotθWith practice these patterns become second nature. Solver AI can verify any trig identity step by step if you want to check your reasoning or see alternative proof paths.