Trig Identities Calculator
Enter a trigonometric expression to simplify it using standard identities.
Use sin(x), cos(x), tan(x) and ^ for powers.
Trigonometric identities are equations true for every angle, such as the Pythagorean identity sin²x + cos²x = 1. This calculator simplifies a trig expression by applying these identities, reducing sin(x)² + cos(x)² to 1 and rewriting other combinations into their simplest equivalent form.
How to Use This Calculator
Enter your trigonometric expression using the standard function names sin, cos and tan with their angle in parentheses, and ^ for powers. The calculator applies trigonometric identities to reduce it to its simplest equivalent form.
Simplifying with identities is a core skill in trigonometry and calculus, turning a complicated-looking expression into something far easier to evaluate, differentiate or integrate.
How the Calculation Works
The calculator recognizes the standard identities — the Pythagorean identities, the quotient identities that define tangent, and the reciprocal identities — and applies them to collapse the expression.
The most famous is sin²x + cos²x = 1, which turns that whole sum into a single number. Others rewrite tangent, secant and cosecant in terms of sine and cosine so that terms can combine and cancel.
sin²x + cos²x = 1; 1 + tan²x = sec²x; tan x = sin x / cos x
Worked Example
Enter sin(x)² + cos(x)². By the Pythagorean identity, this equals exactly 1 for every angle x, so the calculator returns 1.
Another example: tan(x)·cos(x) simplifies to sin(x), because tan(x) is sin(x)/cos(x), and the cos(x) cancels.
Assumptions and Limitations
- Applies standard trig identities to simplify an expression
- Uses radians as the angle unit, the calculus convention
- Results are equivalent for all angles where the expression is defined
- Very unusual expressions may not reduce to a shorter form
Who Uses a Trig Identities Calculator
Trigonometry and precalculus students use this to simplify a trigonometric expression using standard identities. It helps with verifying identities, cleaning up expressions before integration, and checking homework. Enter a trig expression to see it simplified.
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Open calculator →Frequently Asked Questions
What is the most important trig identity?
The Pythagorean identity, sin²x + cos²x = 1, which holds for every angle. It underlies most trig simplification and appears throughout trigonometry and calculus.
How do you simplify a trig expression?
Rewrite everything in terms of sine and cosine, apply the Pythagorean and quotient identities, and cancel or combine terms. This calculator does that automatically and returns the simplest form.
What does tan(x) equal in terms of sine and cosine?
tan(x) = sin(x) ÷ cos(x). This quotient identity lets you convert tangents into sines and cosines so terms can combine, which is often the key step in a simplification.
Does this use degrees or radians?
Radians, the standard unit in trigonometry and calculus. Identities hold regardless of the unit, but function values and any numeric results assume radians.
Can it prove an identity?
It simplifies each side to a canonical form; if both sides reduce to the same thing, they are identical. Simplify each side separately and compare the results.
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