Trignometric Identities:

Pythagorean identities:

The Pythagorean identities come from the Pythagorean theorem and relate sine with cosine, tangent with secant, and cotangent with cosecant.

\(\sin^2\theta + \cos^2\theta = 1\)
\(\tan^2\theta + 1 = \sec^2\theta\)
\(\cot^2\theta + 1 = \csc^2\theta\)

Even-Odd Identities:

The even-odd identities show which trigonometric function is an odd function or an even function.

\(\sin(-\theta) = -\sin\theta\)
\(\csc(-\theta) = -\csc\theta\)
\(\cos(-\theta) = \cos\theta\)
\(\sec(-\theta) = \sec\theta\)
\(\tan(-\theta) = -\tan\theta\)
\(\cot(-\theta) = -\cot\theta\)

Cofunction Identities:

The cofunction identities show the relationship between the sine and the cosine, tangent and cotangent, and secant and cosecant angle measures.

\(\cos(90^{\circ} - \theta) = \sin\theta\)
\(\sin(90^{\circ} - \theta) = \cos\theta\)
\(\tan(90^{\circ} - \theta) = \cot\theta\)
\(\cot(90^{\circ} - \theta) = \tan\theta\)
\(\sec(90^{\circ} - \theta) = \csc\theta\)
\(\csc(90^{\circ} - \theta) = \sec\theta\)

Sum and Difference Identities:

\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)
\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)

Double Angle Identities:

\(\sin(2\theta) = 2 \sin \theta \cos \theta\)
\(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)
\(\cos(2\theta) = 1 - 2\sin^2\theta\)
\(\cos(2\theta) = 2\cos^2\theta - 1\)
\(\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\)

Half-Angle Identities:

\(\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}\)
\(\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}\)

Algebraic Properties of Limits

Assuming that the limits exist,
  1. If \(b\) is a constant, then \(\lim_{{x \to c}}(bf(x)) = b(\lim_{{x \to c}}f(x))\)
  2. \(\lim_{{x \to c}}(f(x) + g(x)) = \lim_{{x \to c}}f(x) + \lim_{{x \to c}}g(x)\)
  3. \(\lim_{{x \to c}}(f(x)g(x)) = (\lim_{{x \to c}}f(x))(\lim_{{x \to c}}g(x))\)
  4. \(\lim_{{x \to c}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)}\), provided \(\lim_{{x \to c}} g(x) \neq 0\)
  5. For any constant k, \(\lim_{{x \to c}} k = k\)
  6. \(\lim_{{x \to c}} x = c\)
  7. \(\lim_{{x \to c}} f(g(x)) = f(\lim_ {{x \to c}} g(x))\)