Identity choice and domain preservation

1. Use the identity without changing the denominator.

   $$\sin^2x/\sin x$$

2. Cancel on the permitted domain.

   $$\sin x,\quad x\ne k\pi,\ k\in\mathbb Z$$

sin x with x≠kπ.

Checks and common pitfalls: The simplified formula alone would conceal the excluded zeros.

1. Eliminate sine squared.

   $$\cos2x=2\cos^2x-1$$

2. Eliminate cosine squared instead.

   $$\cos2x=1-2\sin^2x$$

2cos²x−1 and 1−2sin²x.

Checks and common pitfalls: Different forms serve different known data; neither requires division.

1. The individual tangent values exist.

   $$\tan45^\circ=1$$

2. The target tangent and formula denominator are undefined at the combined angle.

   $$1-1=0,\quad\tan90^\circ\text{ undefined}$$

No real value; the addition formula requires a nonzero denominator.

Checks and common pitfalls: The identity does not authorise division by zero or introduce a real infinity.

1. Write the unit-circle relation.

   $$(\cos x)^2+(\sin x)^2=1$$

2. Order of the sum does not matter.

   $$\sin^2x+\cos^2x=1$$

1.

Checks and common pitfalls: No inverse trigonometric calculation is needed.

1. Select the identity matching both data.

   $$\sin2x=2\sin x\cos x$$

2. Substitute without recovering x.

   $$\sin2x=24/25$$

24/25.

Checks and common pitfalls: The supplied signed values already determine the product.

1. Start from the difference of squares.

   $$\cos2x=\cos^2x-\sin^2x$$

2. Replace the first square.

   $$\cos2x=1-2\sin^2x$$

cos 2x.

Checks and common pitfalls: The factor 2 belongs to the angle in the final form.

1. Evaluate the left side.

   $$\sin60^\circ=\sqrt3/2$$

2. Evaluate the right side and compare.

   $$1/2+1/2=1\ne\sqrt3/2$$

The claim is false.

Checks and common pitfalls: One valid counterexample refutes a purported identity.

1. Write common factors.

   $$\frac{2\sin^2x}{2\sin x\cos x}$$

2. Cancel and retain every original exclusion.

   $$\tan x,\quad x\ne k\pi/2,\ k\in\mathbb Z$$

tan x, excluding x=kπ/2.

Checks and common pitfalls: tan x alone excludes only odd half-turn multiples; the original fraction has additional holes.

1. Expand the sum with the cross terms.

   $$\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$$

2. Insert exact unit-circle values.

   $$=(\sqrt6+\sqrt2)/4$$

(√6+√2)/4.

Checks and common pitfalls: The two cross terms distinguish sine addition from false additivity.

1. The nonzero cosine permits the ratio transformation.

   $$\cos2x=\frac{1-\tan^2x}{1+\tan^2x}$$

2. Substitute the known tangent.

   $$(1-4)/(1+4)=-3/5$$

−3/5.

Checks and common pitfalls: Squaring tangent removes the quadrant ambiguity for this expression.

1. Expand both terms with correct signs.

   $$\sin x\cos y+\cos x\sin y+\sin x\cos y-\cos x\sin y$$

2. Collect identical surviving terms.

   $$2\sin x\cos y$$

The identity holds for all real angles in consistent units.

Checks and common pitfalls: No division is used, so no extra exclusions are introduced.

1. Cross terms cancel in the sum of squares.

   $$u^2+(4\sin x-3\cos x)^2=25$$

2. Nonnegativity bounds u; equality occurs at compatible coordinates.

   $$\sin x=3/5,\quad\cos x=4/5\Rightarrow u=5$$

|u|≤5; equality at sine 3/5, cosine 4/5.

Checks and common pitfalls: This repurposes an identity as an optimisation certificate, including attainability.

1. Verify the equality on the original domain.

   $$(\sin x+\cos x)^2-1=2\sin x\cos x=\sin2x$$

2. Retain the excluded points in the graph.

   $$F(x)=1,\quad x\ne k\pi/2$$

The line y=1 has holes at every x=kπ/2.

Checks and common pitfalls: A simplified expression is an extension if the original exclusions are erased.