- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.
Pythagorean Identities
The second and third Pythagorean identities can be obtained by manipulating the first. The identity [latex]1+{\cot }^{2}\theta ={\csc }^{2}\theta[/latex] is found by rewriting the left side of the equation in terms of sine and cosine.
[latex]\begin{align}1+{\cot }^{2}\theta& =\left(1+\frac{{\cos }^{2}\theta }{{\sin }^{2}\theta }\right)&& \text{Rewrite the left side}. \\ &=\left(\frac{{\sin }^{2}\theta }{{\sin }^{2}\theta }\right)+\left(\frac{{\cos }^{2}\theta }{{\sin }^{2}\theta }\right)&& \text{Write both terms with the common denominator}. \\ &=\frac{{\sin }^{2}\theta +{\cos }^{2}\theta }{{\sin }^{2}\theta } \\ &=\frac{1}{{\sin }^{2}\theta } \\ &={\csc }^{2}\theta \end{align}[/latex]
Similarly, [latex]1+{\tan }^{2}\theta ={\sec }^{2}\theta[/latex] can be obtained by rewriting the left side of this identity in terms of sine and cosine.
[latex]\begin{align}1+{\tan }^{2}\theta &=1+{\left(\frac{\sin \theta }{\cos \theta }\right)}^{2}&& \text{Rewrite left side}. \\ &={\left(\frac{\cos \theta }{\cos \theta }\right)}^{2}+{\left(\frac{\sin \theta }{\cos \theta }\right)}^{2}&& \text{Write both terms with the common denominator}. \\ &=\frac{{\cos }^{2}\theta +{\sin }^{2}\theta }{{\cos }^{2}\theta } \\ &=\frac{1}{{\cos }^{2}\theta } \\ &={\sec }^{2}\theta \end{align}[/latex]
How To: Using the Primary Pythagorean Identity
- Write the identity: [latex]\sin^2\theta + \cos^2\theta = 1[/latex]
- Substitute the known value
- Solve for the unknown function squared
- Take the square root (remember: two possible values)
- Use the quadrant to determine the correct sign
Rewrite [latex]\sec^2\theta[/latex] in terms of sine only.