LEMMA

Trigonometry

Trigonometric identities

ALEKS placement

The Pythagorean identity converts between sine and cosine anywhere.

What this covers

sin2θ+cos2θ=1,tanθ=cosθsinθ​(cosθ=0)
  • Rewrite everything in terms of sine and cosine.
  • Use the Pythagorean identity to trade one for the other.
  • Combine fractions over a common denominator.
  • Keep every original denominator nonzero. Canceling a factor does not remove its restriction.
  • Work on one side only until it matches the other.

Worked example

Worked example

Simplify (1 - cos²θ)/sin θ.

  1. sinθsin2θ​1 - cos²θ = sin²θ.
  2. sinθ,sinθ=0Cancel one factor of sin θ, keeping sin θ ≠ 0 from the original denominator.
Another worked example

Simplify sin θ tan θ + cos θ, preserving its domain.

  1. cosθsin2θ​+cosθ,cosθ=0Replace tangent by sine over cosine, keeping the nonzero cosine restriction.
  2. cosθsin2θ+cos2θ​,cosθ=0Write the second term over the same denominator before adding.
  3. cosθ1​=secθ,cosθ=0The numerator becomes 1 by the Pythagorean identity. The original restriction still applies.

A common mistake

A mistake Lemma catches

From cosθsin2θ​+cosθ

cosθsin2θ​+cosθ=cosθsin2θ+cosθ​cosθsin2θ​+cosθ=cosθsin2θ+cos2θ​

“To put cos θ over the denominator cos θ, multiply its top and bottom by cos θ: cos θ = cos²θ/cos θ. The top becomes sin²θ + cos²θ, which is 1. Check at θ = π/3: the line above is (3/4)/(1/2) + 1/2 = 2, but the struck fraction gives 5/2.”

Try one

Sample problem

Simplify exactly.

sin2(θ)+cos2(θ)

Practice trig identities free

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Where this fits

Before this

After this