Trigonometry & Heights / Distances • Topic 3 of 4

Identities

Three Pythagorean identities do most of the simplifying: sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, and 1 + cot^2 = cosec^2. Rearrange them to swap between functions, and use them to prove or evaluate expressions. A common SSC move: replace sec^2 - tan^2 by 1, or 1 - sin^2 by cos^2, to collapse a long expression.

✅ Solved examples

1. Simplify sec^2 A - tan^2 A.
Identity: equals 1.
2. If sin A = 3/5, find sec A.
cos = 4/5, sec = 1/cos = 5/4.
3. Simplify (1 - cos^2 A).
Equals sin^2 A.
4. cosec^2 A - cot^2 A = ?
Identity: equals 1.

✏️ Practice — try these, take hints as needed

1. Simplify 1 - sin^2 A.
Identity.
—
—
cos^2 A
2. 1 + tan^2 A = ?
Identity.
—
—
sec^2 A
3. cos A = 5/13 -> cosec A?
sin = 12/13.
1/sin.
—
13/12
4. sec^2 30 - tan^2 30?
Identity = 1.
—
—
1
5. Simplify sin A cosec A.
cosec = 1/sin.
Product.
—
1

📝 Topic test — 8 questions

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