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Pythagorean Trigonometric Identities

A trigonometric identity in one variable is an equality that involves trigonometric functions and is true for all values of the variable for which both sides of the equality are defined.

Recall the Pythagorean theorem that relates the lengths of the sides of a right triangle:

\[{a^2} + {b^2} = {c^2},\]

where a, b are the lengths of the triangle's legs and c is the length of its hypotenuse.

A right triangle illustrating the Pythagorean theorem
Figure 1.

The sine and cosine functions of an acute angle \(\alpha\) in the right triangle are expressed in terms of the sides \(a,b, \text{and } c:\)

\[\sin \alpha = \frac{a}{c},\;\;\cos \alpha = \frac{b}{c}.\]

By squaring and adding these equations, we get the famous relationship

\[{\sin ^2}\alpha + {\cos ^2}\alpha = \frac{{{a^2}}}{{{c^2}}} + \frac{{{b^2}}}{{{c^2}}} = \frac{{{a^2} + {b^2}}}{{{c^2}}} = \frac{{{c^2}}}{{{c^2}}} = 1,\]

which is called the Pythagorean trigonometric identity. Thus, for any angle \(\alpha\) in the range \(0 \le \alpha \le \frac{\pi}{2},\) we have

\[\sin^2\alpha + \cos^2\alpha = 1.\]

Later we can see that this identity is valid for any real value of \(\alpha.\)

Assuming \(\cos \alpha \ne 0\) (so that \(\alpha \ne \frac{\pi }{2}\)), we can divide both sides of the equation by \({\cos^2}\alpha.\) This yields:

\[\require{cancel}\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} + \frac{\cancel{{{\cos }^2}\alpha }}{\cancel{{{\cos }^2}\alpha }} = \frac{1}{{{{\cos }^2}\alpha }},\]


\[\tan^2\alpha + 1 = \sec^2\alpha.\]

This identity relates the tangent and secant functions of an angle.

Similarly, if \(\sin \alpha \ne 0,\) that is \(\alpha \ne 0,\) we can divide the initial Pythagorean trigonometric identity \({\sin^2}\alpha + {\cos^2}\alpha = 1\) by \({\sin^2}\alpha\) to obtain an equation relating the cotangent and cosecant functions:

\[\frac{\cancel{{{\sin }^2}\alpha }}{\cancel{{{\sin }^2}\alpha }} + \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} = \frac{1}{{{{\sin }^2}\alpha }},\]


\[\cot^2\alpha + 1 = \csc^2\alpha.\]

The last two formulas are also called Pythagorean trigonometric identities.

These identities allow us to convert between the trigonometric functions of an angle, without knowing the angle itself, and simplify trig expressions.

See solved problems on Page 2.

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