Compound, double-angle and reciprocal identities

Reciprocal functions are ratios, not inverse angles

Secant requires a nonzero cosine; cosecant and cotangent require a nonzero sine. Inverse sine denotes an angle, whereas cosecant is a reciprocal ratio

If sin A=−5/13 and cos A=12/13, find csc A.

Find cot A for the angle with these ratios

Compound sine and cosine

Use the same sign in the sine formula and the opposite sign in the cosine cross term. These identities hold for all real A and B, provided angle units are used consistently

Worked exact sum

The sum can be in a different quadrant from either input; calculate the signs

Find sin(A−B) using these ratios

Find cos(A−B) using these ratios

Derive the compound identities, ungraded

Place unit-circle points P=(cos A,sin A) and Q=(cos B,sin B). Calculate their squared distance in coordinates and with the cosine rule. Derive cos(A−B), then derive cos(A+B) and both sine formulas. Written proof is ungraded

Model: compare two distance expressions

Calculate the squared distance from the coordinates and from the cosine rule. The smaller geometric separation has the same cosine as the angle difference. Equate the distances, then replace B by its negative and use the even and odd properties. Use the complementary-angle identity for sine sum, then replace B by its negative for sine difference

Double an angle by setting B=A

The compound identities give sin2A=2sin A cos A and cos2A=cos²A−sin²A. Substitute sin²A+cos²A=1 to obtain the other cosine forms. Choose the form that uses the information supplied

If sin A=5/13 in quadrant II, find sin2A

Find cos2A for an angle in quadrant II with

Find tan2A for an angle in quadrant II with

Tangent needs extra domain checks

Divide the sine and cosine compound identities by cos A cos B. This step needs both input cosines nonzero, and the final denominator must also be nonzero. For double angles, tan2A=2tan A/(1−tan²A), with the same restrictions

If tan A=3/4 and tan B=5/12, find tan(A+B)

For A=90° and B=45°, which statement is correct?

Reciprocal Pythagorean identities

Divide sin²A+cos²A=1 by cos²A, or by sin²A. Each division imposes a restriction, which must remain attached to the resulting identity

If cot A=−4/3, find csc²A.

On what domain is (sec²A−1)/tan A equal to tan A?

Prove an identity with its original domain, ungraded

Prove (1−cos2A)/sin2A=tan A and state precisely which angles the original left side excludes. Explain why cancellation cannot put excluded angles back. This proof is ungraded

Model: factor, then cancel on the stated domain

Use the double-angle formulas and keep the original denominator restriction when cancelling. At zero, the simplified tangent exists but the original quotient does not

Exact values beyond the basic angles

Use 15°=45°−30° and 75°=45°+30°. For example, sin15°=(√6−√2)/4 and cos15°=(√6+√2)/4. Keep radicals exact; a decimal check does not establish an exact identity

Write sin75°=u√6+v√2. Find v.

Apply identities to an equation, ungraded

Solve sin2x=sin x for 0≤x<2π without losing solutions by division. Check all candidates in the original equation

Model: factor instead of dividing

All four values satisfy the original equation. Dividing by sine at the start would lose two solutions