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There are three different versions of the Pythagorean Identities. We will use $\sin ^2(\theta)+\cos ^2(\theta)=1$ to obtain the other two versions.


AWe assume that $\cos (\theta)$ is not $0$. Divide both sides of the identity $\sin ^2(\theta)+\cos ^2(\theta)=1$ by $\cos ^2(\theta)$. What do we have?

Write the result from part a in the following expression:

$\left(\frac{\sin ^2(\theta)}{\cos ^2(\theta)}\right)^2+1=\left(\frac{\cos ^2}{\cos ^2}\right)^2$



We can simplify the expression in part b by recognizing the equivalent formulae to the terms inside the parentheses.


BWe assume that $\sin (\theta)$ is not $0$. Divide both sides of the identity $\sin ^2(\theta)+\cos ^2(\theta)=1$ by $\sin ^2(\theta)$. What do we have?

Write the result from part a in the following expression:

$(-)^2+1=(\square)^2$



We can simplify the expression in part b by recognizing the equivalent formulae to the terms inside the parentheses.

Check your results in parts d and f with the formula sheet.

Verified Answer :

  • Start with the base identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).
  • Divide by \(\cos^2(\theta)\) to derive \(\tan^2(\theta) + 1 = \sec^2(\theta)\).
  • Divide by \(\sin^2(\theta)\) to derive \(1 + \cot^2(\theta) = \csc^2(\theta)\).
  • The complete set of Pythagorean identities is \(\boxed{\sin^2(\theta) + \cos^2(\theta) = 1, \tan^2(\theta) + 1 = \sec^2(\theta), 1 + \cot^2(\theta) = \csc^2(\theta)}\).

Explanation

  1. Analysis of the Fundamental Identity

We start with the fundamental Pythagorean identity:
\[\sin^2(\theta) + \cos^2(\theta) = 1\]
This identity relates the sine and cosine of an angle \(\theta\). We aim to derive two additional identities from this one by dividing by \(\cos^2(\theta)\) and \(\sin^2(\theta)\), assuming these values are non-zero.

  1. Deriving the First Identity (Tan-Sec)

Assuming \(\cos(\theta) eq 0\), we divide both sides of \(\sin^2(\theta) + \cos^2(\theta) = 1\) by \(\cos^2(\theta)\):
\[\frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}\]
This gives us the identity:
\[\left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2 + 1 = \frac{1}{\cos^2(\theta)}\]
Given the definitions \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\), this expression simplifies to:
\[\tan^2(\theta) + 1 = \sec^2(\theta)\]

  1. Deriving the Second Identity (Cot-Csc)

Assuming \(\sin(\theta) eq 0\), we divide both sides of \(\sin^2(\theta) + \cos^2(\theta) = 1\) by \(\sin^2(\theta)\):
\[\frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)}\]
This results in:
\[1 + \left(\frac{\cos(\theta)}{\sin(\theta)}\right)^2 = \frac{1}{\sin^2(\theta)}\]
Given the definitions \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\), this expression simplifies to:
\[1 + \cot^2(\theta) = \csc^2(\theta)\]

  1. Conclusion of Derivations

We have derived the following three Pythagorean identities:

  1. \(\sin^2(\theta) + \cos^2(\theta) = 1\)
  2. \(\tan^2(\theta) + 1 = \sec^2(\theta)\)
  3. \(1 + \cot^2(\theta) = \csc^2(\theta)\)

These identities are fundamental in trigonometry for simplifying expressions and solving equations involving trigonometric functions. The derivations confirm that \(\tan^2(\theta) + 1 = \sec^2(\theta)\) and \(1 + \cot^2(\theta) = \csc^2(\theta)\) are consistent with standard mathematical references. Final identities are: \(\boxed{\sin^2(\theta) + \cos^2(\theta) = 1, \tan^2(\theta) + 1 = \sec^2(\theta), 1 + \cot^2(\theta) = \csc^2(\theta)}\).

Examples

Trigonometric identities are widely used in engineering, physics, and computer graphics to simplify complex wave equations or coordinate transformations. For instance, if you are designing a rotating engine part or programming a character movement in a video game, you often need to calculate distances or angles using sine and cosine. Knowing these identities allows you to replace complex trigonometric expressions with simpler, equivalent forms, making your calculations much faster and more accurate.

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