Double Angle: Cosine

This identity gives the cosine of an angle doubled, 2θ, in terms of the sine and cosine of θ itself. It comes from the cosine addition rule, since cos(θ+θ) expands to cos θ cos θ − sin θ sin θ, which simplifies to cos²θ − sin²θ. The same relationship can be rearranged into equivalent forms such as 2cos²θ−1 or 1−2sin²θ, depending on which trig ratio is easier to work with.

Use this when you need to simplify expressions involving cos of a doubled angle, or when you know either sin θ or cos θ and want to find cos 2θ without directly finding the angle. It is also common in trigonometric proofs, equation solving, and geometry problems involving repeated angles.

• Using cos 2θ = 2cos²θ − 2sin²θ instead of the correct identity
• Forgetting that the double-angle formula is for 2θ, not θ
• Substituting sin θ or cos θ without squaring the value first