Trig identities and proofs

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What this lesson covers

  • sin²x + cos²x = 1.
  • 1 + tan²x = sec²x.
  • Start from one side and finish at the other.
Key idea

The facts you need

  • tan x = sin x ÷ cos x.
  • sec x = 1 ÷ cos x, cosec x = 1 ÷ sin x and cot x = cos x ÷ sin x.
  • sin²x + cos²x = 1 and 1 + tan²x = sec²x.
Worked example

Simplify (1 − cos²x) ÷ sin x

  • Replace the top with a single term.
  1. = sin²x ÷ sin x
  2. = sin x
Worked example

Prove tan x + cot x = sec x cosec x

  • Work from the left side, and write everything in sine and cosine.
  1. LHS = sin x/cos x + cos x/sin x
  2. = (sin²x + cos²x) ÷ (sin x cos x)
  3. = 1 ÷ (sin x cos x)
  4. = sec x cosec x = RHS

Common mistakes

  • Moving terms across the equals sign in a proof.
    Work on one side only.
  • Cancelling sin from sin x + 1 ÷ sin x.
    Only common factors cancel.

Your turn

  • 1. Simplify sin x cot x.
  • 2. Prove (1 + tan²x) cos²x = 1.
Show answer

1. cos x 2. LHS = sec²x cos²x = 1

Summary

What to take away

  • When stuck, write it all in sine and cosine.
  • A proof runs from one side to the other.

More in AS 91575

Trig functions and radiansNotesTrig identities and proofsNotesCompound and double-angle formulaeNotesTrig equations and general solutionsNotesModelling periodic situationsNotes

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