(1-cos x)(1+cos x)=sin2x and (1-sin x)(1+sin x)=cos2x. These are factorised Pythagorean identities. A conjugate factor changes a difference of squares into a single squared ratio. Combining fractions first often creates the same pair of factors automatically. 1. Record every original denominator. 2. Choose a common denominator or multiply numerator and denominator by a conjugate. 3. Apply the appropriate Pythagorean identity. 4. Cancel only nonzero factors and identify the required side. Example: 1/(1-cos x)+1/(1+cos x)=2/(1-cos2x)=2/sin2x. For x=pi/3, each side is 8/3. The original denominators require sin x 0. Exam check: cancelling a factor does not enlarge the identity domain. For example (1-cos x)/sin x is still undefined at x=0, even if an equivalent expression has a value there.

\((1-\cos x)(1+\cos x)=\sin^2x\) and \((1-\sin x)(1+\sin x)=\cos^2x\). These are factorised Pythagorean identities.

A conjugate factor changes a difference of squares into a single squared ratio. Combining fractions first often creates the same pair of factors automatically.

1. Record every original denominator. 2. Choose a common denominator or multiply numerator and denominator by a conjugate. 3. Apply the appropriate Pythagorean identity. 4. Cancel only nonzero factors and identify the required side.

Example: \(\frac{1}{1-\cos x}+\frac{1}{1+\cos x}=\frac{2}{1-\cos^2x}=\frac{2}{\sin^2x}\). For \(x=\frac{\pi}{3}\), each side is \(\frac{8}{3}\). The original denominators require \(\sin x\ne0\).

Exam check: cancelling a factor does not enlarge the identity domain. For example \(\frac{1-\cos x}{\sin x}\) is still undefined at \(x=0\), even if an equivalent expression has a value there.

Fractions and conjugate factors

Fractions and conjugate factors