Prove identities on their shared domains

Try

A point on the unit circle has coordinates (cos θ,sin θ). Its squared distance from the origin is 1. Write that geometric statement as cos²θ+sin²θ=1. This single identity can replace a sum of two complicated squared expressions with a constant, but only when they use the same angle.

Predict

Predict whether sin²a+cos²b must equal 1 when a and b differ. Try a=0 and b=π/2: the sum is zero. Matching notation matters. Also predict whether knowing sin θ=3/5 determines cosine’s sign without a quadrant restriction.

Experiment

Use the identity to complete two possible unit-circle points with vertical coordinate 3/5. Their horizontal coordinates are 4/5 and −4/5. Sketch both. The squared identity determines magnitude, while the location determines sign.

Next divide every term of sin²θ+cos²θ=1 by cos²θ. Record the necessary condition cos θ≠0 before writing tan²θ+1=sec²θ. Repeat by dividing by sin²θ to obtain 1+cot²θ=csc²θ, with sin θ≠0. Each division creates its own domain condition.

Understand

An identity asserts equality for every input on the stated shared domain. An equation may hold only for selected inputs. Checking a few angles can suggest an identity but cannot prove its universal claim; one valid counterexample is enough to refute it.

For (1−cos²x)/sin x, replace the numerator by sin²x and cancel one sine factor. The result is sin x where sin x is nonzero. At x=0, the original is 0/0 and undefined, even though the simplified expression is defined. A complete simplification includes the retained restriction.

A useful proof method transforms one side using known identities until it becomes the other. State every division condition. Avoid beginning with the desired equality and applying a nonreversible operation, such as squaring, then treating a true final statement as proof of the original. Two unequal quantities can have equal squares.

The common false rule sin(a+b)=sin a+sin b illustrates the difference between a function and a linear operation. At a=b=π/2, the left side is zero and the right side is two. Trigonometric functions are not generally additive. Addition formulas require cross terms involving both sine and cosine; the next lesson derives them from rotation.

Apply

For numeric practice, decide whether the task asks for a square or a signed function value. For written practice, include the domain as part of the answer. When disproving an identity, show both exact values at an allowed input; an undefined input is not a counterexample to an identity expressly limited to its shared domain.

Continue

Angle addition transforms coordinates in a structured way. That geometry explains the missing cross terms and provides exact values for angles such as π/12 without decimal approximations.