Textbook Question
Match each expression in Column I with its value in Column II.
10. cos 67.5°
707
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Multiple Choice
Which of the following is not a variation of a Pythagorean identity?
A
B
C
D
Verified step by step guidance1
Recall the fundamental Pythagorean identity: \(\sin^{2}x + \cos^{2}x = 1\). This is the base from which other variations are derived.
Use the definitions of tangent and secant in terms of sine and cosine: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). Substitute these into the identity to derive \(\tan^{2}x + 1 = \sec^{2}x\).
Similarly, use the definitions of cotangent and cosecant: \(\cot x = \frac{\cos x}{\sin x}\) and \(\csc x = \frac{1}{\sin x}\). Substitute these into the identity to get \(\cot^{2}x + 1 = \csc^{2}x\).
Check each given equation against these known identities. The first three equations correspond to these variations, while the last one, \(\sin^{2}x + \tan^{2}x = 1\), does not follow from the fundamental identity or its standard variations.
Conclude that the equation \(\sin^{2}x + \tan^{2}x = 1\) is not a variation of the Pythagorean identity because it cannot be derived from the basic identity or the definitions of tangent and secant.
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