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3.3 Trigonometric functions of sum and difference of two angles > No. 14

Prove the given identity by using basic trigonometric formulas.

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Explanation:

Trigonometry

Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

http://en.wikipedia.org/wiki/Trigonometry

Symmetry

When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:

Reflected in $θ = 0$	Reflected in $θ = π / 4$ (co-function identities)	Reflected in $θ = π / 2$
$Description: Description: Description: \begin{align} \sin(-\theta) &= -\sin \theta \\ \cos(-\theta) &= +\cos \theta \\ \tan(-\theta) &= -\tan \theta \\ \csc(-\theta) &= -\csc \theta \\ \sec(-\theta) &= +\sec \theta \\ \cot(-\theta) &= -\cot \theta \end{align}$	$Description: Description: Description: \begin{align} \sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\ \cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\ \tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\ \csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\ \sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\ \cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta \end{align}$	$Description: Description: Description: \begin{align} \sin(\pi - \theta) &= +\sin \theta \\ \cos(\pi - \theta) &= -\cos \theta \\ \tan(\pi - \theta) &= -\tan \theta \\ \csc(\pi - \theta) &= +\csc \theta \\ \sec(\pi - \theta) &= -\sec \theta \\ \cot(\pi - \theta) &= -\cot \theta \\ \end{align}$

Double-angle formula
$Description: Description: Description: \begin{align} \sin 2\theta &= 2 \sin \theta \cos \theta \ \\ &= \frac{2 \tan \theta} {1 + \tan^2 \theta} \end{align}$	$Description: Description: Description: \begin{align} \cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ &= 2 \cos^2 \theta - 1 \\ &= 1 - 2 \sin^2 \theta \\ &= \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta} \end{align}$	$Description: Description: Description: \tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\!$	$Description: Description: Description: \cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\!$

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.

(Our solved example in mathguru.com also uses the below concept. This is our own explanation, it is not taken from Wikipedia.)

TRIGNOMETRIC FORMULA USED

sin A + sin B = 2 sin () cos ()