7.4 Proving Trigonometric Identities
Chapter
Chapter 7
Section
7.4
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Solutions 40 Videos

John claims that \sin x = \cos x is an identity, since \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}. Use a counterexample to disprove his claim.

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1.13mins
Q1

(a) Use a graphing calculator to graph f(x) = \sin x and g(x) = \tan x \cos x for -2\pi \leq x\leq 2\pi.

(b) Write a trigonometric identity based on your graphs.

(c) Simplify one side of your identity to prove it is true.

(d) This identity is true for all real numbers, except where \cos x = 0. Explain.

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1.27mins
Q2

Prove algebraically that the expressions are equivalent.

\displaystyle \sin x\cot x = \cos x

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0.50mins
Q4a

Prove algebraically that the expressions are equivalent.

\displaystyle 1- 2\sin^2x = 2\cos^2x - 1

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0.43mins
Q4b

Prove algebraically that the expressions are equivalent.

\displaystyle (\sin x +\cos x)^2 = 1+ 2\sin x\cos x

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0.57mins
Q4c

Prove algebraically that the expressions you matched in question 3 are equivalent.

\displaystyle \sin^2x \cos^2x + \tan^2x

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1.07mins
Q4d

Give a counterexample to show that each equation is not an identity.

\displaystyle \cos x =\frac{1}{\cos x}

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0.26mins
Q5a

Give a counterexample to show that each equation is not an identity.

\displaystyle 1 - \tan^2x = \sec^2x

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1.56mins
Q5b

Give a counterexample to show that each equation is not an identity.

\displaystyle \sin(x + y) = \cos x \cos y + \sin x \sin y

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0.46mins
Q5c

Give a counterexample to show that each equation is not an identity.

\displaystyle \cos 2x = 1 +2\sin^2x

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2.04mins
Q5d

Graph the expression \displaystyle \frac{1 - \tan^2x}{1 + \tan^2x}, and make a conjecture about another expression that is equivalent to this expression.

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0.41mins
Q6

Prove that \displaystyle \frac{1 - \tan^2x}{1 + \tan^2x} = \cos 2x.

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1.26mins
Q7

Prove that \displaystyle \frac{1 + \tan x}{1 - \tan x} = \frac{1 - \tan x}{\cot x - 1}.

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3.33mins
Q8

Prove each identity.

\displaystyle \frac{\cos^2\theta -\sin^2\theta}{\cos^2\theta + \sin\theta\cos\theta} = 1-\tan\theta

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1.05mins
Q9a

Prove each identity.

\displaystyle \tan^2x - \sin^2x = \sin^2x \tan^2x

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1.11mins
Q9b

Prove each identity.

\displaystyle \tan^2x -\cos^2x =\frac{1}{\cos^2x} - 1 -\cos^2x

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1.07mins
Q9c

Prove each identity.

\displaystyle \frac{1}{1 +\cos \theta} + \frac{1}{1 - \cos \theta} = \frac{2}{\sin^2\theta}

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1.40mins
Q9d

Prove each identity.

\displaystyle \cos x \tan^3x = \sin x \tan^2x

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1.12mins
Q10a

Prove each identity.

\displaystyle \sin^2 \theta + \cos^4\theta = \cos^2 \theta + \sin^4\theta

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2.14mins
Q10b

Prove each identity.

\displaystyle \displaystyle (\sin x + \cos x)(\frac{\tan^2x + 1}{\tan x}) = \frac{1}{\cos x} + \frac{1}{\sin x}

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2.28mins
Q10c

Prove each identity.

\displaystyle \tan^2 \beta + \cos^2 \beta + \sin^2 \beta = \frac{1}{\cos^2 \beta}

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0.53mins
Q10d

Prove each identity.

\displaystyle \sin(\frac{\pi}{4} + x) + \sin(\frac{\pi}{4} - x) = \sqrt{2}\cos x

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0.56mins
Q10e

Prove each identity.

\displaystyle \sin(\frac{\pi}{2} - x)\cot(\frac{\pi}{2} + x) = -\sin x

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3.03mins
Q10f

Prove each identity.

\displaystyle \frac{\cos 2x + 1}{\sin 2x} = \cot x

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0.56mins
Q11a

Prove each identity.

\displaystyle \frac{\sin 2x }{1- \cos 2x} = \cot x

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2.06mins
Q11b

Prove each identity.

\displaystyle (\sin x + \cos x)^2 = 1 + \sin 2x

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0.26mins
Q11c

Prove each identity.

\displaystyle \cos^4x - \sin^4x = \cos 2x

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1.27mins
Q11d

Prove each identity.

\displaystyle \cot x - \tan x = 2\cot 2x

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2.36mins
Q11e

Prove each identity.

\displaystyle \cot x + \tan x = 2\csc 2x

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2.06mins
Q11f

Prove each identity.

\displaystyle \frac{1 + \tan x}{1 - \tan x} = \tan(x + \frac{\pi}{4})

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0.37mins
Q11g

Prove each identity.

\displaystyle \csc 2x + \cot 2x = \cot x

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1.42mins
Q11h

Prove the identity.

\displaystyle \frac{2\tan x}{1 + \tan^2x} = \sin 2x

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1.04mins
Q11i

Prove each identity.

\displaystyle \sec 2x = \frac{\csc x}{\csc x - 2\sin x}

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2.45mins
Q11j

Prove each identity.

\displaystyle \csc 2\theta = \frac{1}{2}(\sec \theta)(\csc \theta)

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0.53mins
Q11k

Prove each identity.

\displaystyle \sec t = \frac{\sin 2t}{\sin t} - \frac{\cos 2t}{\cos t}

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1.52mins
Q11l

Graph the expression \displaystyle \frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} , and make a conjecture about another expression that is equivalent to this expression.

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0.28mins
Q12

Prove \displaystyle \frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \tan x .

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0.58mins
Q13

Each of the following expressions can be written in the form a \sin 2x + b \cos 2x + c. Determine the values of a, b, and c.

\displaystyle 2\cos^2x + 4\sin x \cos x

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1.28mins
Q16a

Each of the following expressions can be written in the form a \sin 2x + b \cos 2x + c. Determine the values of a, b, and c.

\displaystyle - 2\sin x \cos x - 4\sin^2x

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3.00mins
Q16b

Express 8 \cos^4x in the form a\cos 4x + b\cos 2x + c. State the values of the constants a, b, and c.

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4.30mins
Q17