Trigonometric Identities

• Some important relationships between the trigonometric ratios can be established using Pythagoras' Theorem: 
  \[\sin^2 \theta+ \cos^2 \theta = \frac{a^{2}}{c^{2}} + \frac{b^{2}}{c^{2}} = \frac{a^{2}+b^{2}}{c^{2}}= \frac{c^{2}}{c^{2}} =1\] 
• Dividing through by \(\cos^{2}\theta\) gives:
  \begin{eqnarray*}
  \sin^2 \theta+ \cos^2 \theta &=& 1 \\
  \frac{\sin^{2}\theta}{\cos^{2}\theta} + \frac{\cos^{2}\theta}{\cos^{2}\theta} &=& \frac{1}{\cos^{2}\theta} \\
  \tan^{2}\theta + 1 &=& \sec^{2}\theta 
  \end{eqnarray*} 
• Dividing through by \(\sin^{2}\theta\) gives:
  \begin{eqnarray*}
  \sin^2 \theta+ \cos^2 \theta &=& 1 \\
  \frac{\sin^{2}\theta}{\sin^{2}\theta} + \frac{\cos^{2}\theta}{\sin^{2}\theta} &=& \frac{1}{\sin^{2}\theta} \\
  1 + \cot^{2}\theta &=& \csc^{2}\theta
  \end{eqnarray*}

For example: simplify \( (\sin x +\cos x)^{2} + (\sin x -\cos x)^{2} \) 
\begin{eqnarray*}
&& (\sin x +\cos x)^{2} + (\sin x -\cos x)^{2} \\
&=& \sin^{2}x +\cancel{ 2 \sin x\cos x} + \cos^{2}x \\
&& + \sin^{2}x -\cancel{2\sin x\cos x} + \cos^{2}x \\
&=& 2(\sin^{2}x+ \cos^{2}x ) \\
&=& 2 \times 1 = 2 
\end{eqnarray*}