Textbook Question
Use a half-angle identity to find each exact value.
sin 195°
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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 \(\sec x = \frac{1}{\cos x}\). From these, derive the identity \(\tan^{2}x + 1 = \sec^{2}x\) by dividing the fundamental identity by \(\cos^{2}x\).
Similarly, use the definitions of cotangent and cosecant: \(\cot x = \frac{\cos x}{\sin x}\) and \(\csc x = \frac{1}{\sin x}\). Dividing the fundamental identity by \(\sin^{2}x\) gives \(\cot^{2}x + 1 = \csc^{2}x\).
Examine the given options and check if each can be derived from the fundamental identity or its variations by algebraic manipulation and substitution of trigonometric definitions.
Identify the expression \(\sin^{2}x + \tan^{2}x = 1\) as not fitting the pattern of Pythagorean identities because it mixes sine squared and tangent squared without a direct algebraic derivation from the fundamental identity.
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