Trigonometric identities

The two identities

• tan θ = sin θ / cos θ
• sin² θ + cos² θ = 1

Rearranged: sin² θ = 1 - cos² θ and cos² θ = 1 - sin² θ.

Finding one ratio from another

If sin θ = 3/5, then cos² θ = 1 - 9/25 = 16/25, so cos θ = ±4/5. The quadrant decides the sign: for an obtuse angle, cos θ is negative.

Proving identities

Start with the more complicated side and use the identities to turn it into the other side. Do not work on both sides at once.

Worked example

sin θ = 3/5 and θ is obtuse. Find cos θ and tan θ.

1. cos² θ = 1 - 9/25 = 16/25
2. θ is obtuse, so cos θ < 0: cos θ = -4/5
3. tan θ = (3/5)/(-4/5)

Answer: cos θ = -4/5, tan θ = -3/4

Check yourself

1. sin θ = 5/13 and θ is acute. Find cos θ. Give your answer as a fraction in its simplest form.
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   12/13. cos² θ = 1 - 25/169 = 144/169, so cos θ = 12/13 (positive, as θ is acute).

2. sin θ = 5/13 and θ is acute. Find tan θ. Give your answer as a fraction in its simplest form.
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   5/12. tan θ = (5/13)/(12/13) = 5/12.

3. cos θ = -1/3. Find sin² θ. Give your answer as a fraction in its simplest form.
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   8/9. sin² θ = 1 - 1/9 = 8/9.

4. Simplify (1 - cos² x)/sin x.
   • cos x
   • 1
   • sin x
   • tan x
   Show answer

   sin x. 1 - cos² x = sin² x, and sin² x / sin x = sin x.

5. Simplify sin x / tan x.
   • cos x
   • sin² x
   • 1/cos x
   • tan x
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   cos x. sin x / (sin x / cos x) = cos x.

6. Which expression is identical to (sin x + cos x)²?
   • 1
   • 1 + sin x cos x
   • 1 + 2 sin x cos x
   • 2
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   1 + 2 sin x cos x. sin² x + 2 sin x cos x + cos² x = 1 + 2 sin x cos x.

7. sin x = 2 cos x. Find tan x.
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   2. Divide by cos x: tan x = 2.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.