4.6 Trig Identities
Chapter
Chapter 4
Section
4.6
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Solutions 50 Videos

Using Pythagorean identities, state an equivalent expression.

\displaystyle \sin^2 \theta

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Q1a

Using Pythagorean identities, state an equivalent expression.

\displaystyle \tan^2 \theta

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Q1b

Using Pythagorean identities, state an equivalent expression.

\displaystyle \sec^2 \theta

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Q1c

Using Pythagorean identities, state an equivalent expression.

\displaystyle 1 - \sin^2 \theta

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Q1d

Using Pythagorean identities, state an equivalent expression.

\displaystyle 1 - \csc^2 \theta

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Q1e

Using Pythagorean identities, state an equivalent expression.

\displaystyle \cot^2\theta - \csc^2\theta

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Q1f

Using Pythagorean identities, state an equivalent expression.

\displaystyle -\frac{1}{\csc^2\theta}

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Q1g

Express each expression in a simpler form.

\displaystyle \cos \theta \sec \theta

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Q2a

Express each expression in a simpler form.

\displaystyle \tan \theta \cos \theta

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Q2b

Express each expression in a simpler form.

\displaystyle \tan \theta + \cot \theta

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Q2c

Express each expression in a simpler form.

\displaystyle \sqrt{1 -\cos^2\theta}

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Q2d

Express each expression in a simpler form.

\displaystyle \tan^2\theta -\sec^2 \theta

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Q2e

Express each expression in a simpler form.

\displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta}

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Q2f

Factor to simplify each expression.

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

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Q3a

Factor to simplify each expression.

\displaystyle (\sec \theta)^2(\sin \theta)^2 + (\sin \theta)^2

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Q3b

Factor to simplify each expression.

\displaystyle 4\cos^2\theta + 8 \cos \theta \sin \theta + 4\sin^2 \theta

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Q3c

Factor to simplify each expression.

\displaystyle \sin^4\theta - \cos^4 \theta

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Q3d

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \tan \theta \cos \tan \theta =\sin \tan \theta

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Q4a

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \cot \theta \sec \theta = \csc \theta

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Q4b

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \frac{1 + \cot^2\theta}{\csc^2\theta} = 1

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Q4c

Prove that \displaystyle \cos \theta \cot \theta = \frac{1}{\sin \theta} -\sin \theta

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Q5

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

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Q7

Prove that \displaystyle \frac{1 + \cot \theta}{\csc \theta} = \sin \theta + \cos \theta

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Q8

Prove that \displaystyle \frac{\tan^2\theta}{1 + \tan^2\theta} = \sin^2\theta

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Q9

Prove each identity.

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

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Q11a

Prove each identity.

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

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Q11b

Prove each identity.

\displaystyle \frac{1 + \sec \theta}{\sec \theta - 1} = \frac{1 + \cos \theta}{1 -\cos \theta}

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Q11c

Prove each identity.

\displaystyle \frac{1 -\sin \theta}{1 + \sin \theta} = \frac{\csc \theta -1}{\csc \theta + 1}

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Q11d

Simplify

\displaystyle \sin^2\theta + \frac{1 + \cot^2\theta}{1 + \tan^2\theta} + \cos^2\theta .

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Q12

Prove that 2\sin(-\theta) - \cot\theta \sin\theta \cos\theta = (\sin \theta -1)^2 -2

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Q13

Explain why the identity

\displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} = \frac{2}{\cos \theta} is not true for \theta = 90^o and \theta= 270^o.

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Q15

Prove that

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

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Q16

Prove that

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

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Q17

Prove that

\displaystyle \frac{7\sin \theta + 5\cos \theta}{\cos \theta \sin \theta} = 7 \sec \theta + 5\csc \theta .

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Q18

Prove that

\displaystyle (\csc \theta - \cot \theta)^2 = \frac{1 -\cos \theta}{1 + \cos \theta} .

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Q19

Prove that

\displaystyle \sec^2\theta + \csc^2\theta = \frac{\csc^2\theta}{\cos^2\theta} .

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Q20

Prove that

\displaystyle \sec\theta - \frac{\sin \theta}{\cot \theta} = \frac{1}{\sec \theta} .

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Q21

Prove that

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

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Q22

Prove that

\displaystyle \frac{\tan \theta - \sin \theta}{\sin^3\theta} = \frac{\sec \theta}{1 + \cos \theta} .

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Q23

Prove that

\displaystyle \frac{1 + \sin \theta}{\tan \theta} = \cos \theta + \cot \theta .

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Q24

Prove that

\displaystyle (\tan \theta + \cot \theta)^2 = \sec^2\theta +\frac{1}{\sin^2\theta} .

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Q25

Use a graphing calculator to graph each side to determine if the equation appears to be an identity.

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

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Q26a

Use a graphing calculator to graph each side to determine if the equation appears to be an identity.

\displaystyle \cos x (\cos x -\sec x) = -\sin^2x

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Q26b

Prove that

\displaystyle (\sin \theta + \cos \theta)^2 = \frac{2 + \sec \theta \csc \theta}{\sec \theta \csc \theta}

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Q28

Prove that

\displaystyle \frac{\cot \theta}{\csc \theta -1} = 2\sec \theta

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Q29

Prove that

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

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Q30

Prove that

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

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Q31

Prove that

\displaystyle \tan \theta \sin \theta + \cos \theta - \sec \theta + 1 = \sec^2\theta \cos^2 \theta

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Q32

Prove that

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

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Q33

Prove that \tan^2\theta(1 + \cot^2\theta) = \sec^2\theta

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Q34
Lectures 10 Videos

Introduction to Trig Ratios and to Identity


Trig Ratios

\sin \theta = \dfrac{opp}{hyp}

\cos \theta = \dfrac{adj}{hyp}

\tan \theta = \dfrac{opp}{adj}


Reciprocal Identities

\csc \theta = \dfrac{1}{\sin \theta}

\sec \theta = \dfrac{1}{\cos \theta}

\cot \theta = \dfrac{1}{\tan \theta}


Pythagorean Identities

\sin^2 \theta + \cos^2 \theta = 1

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7.22mins
1 Introduction to Trig Ratios and to Identity

Simplifying \cot x \sin x

ex Simplify cot \theta \times \sin \theta.

= \dfrac{\cos \theta}{\sin \theta} \times \sin \theta

= \cos \theta

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0.33mins
2 Simplifying cotx sin x

Simplifying Trig Expression 1+\tan^2x

ex 1+\tan^2x

= 1+ (\dfrac{\sin \theta}{\cos \theta})^2

= 1 + \dfrac{\sin^2 \theta}{\cos^2 \theta}

= \dfrac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}

= \dfrac{1}{\cos^2 \theta}

= \sec^2 \theta

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1.14mins
3 Simplifying Trig Expression 1+tan^2x

Simplifying Trig Expression \dfrac{1 + \sin \theta}{\cos^2 \theta}

ex Simplify \dfrac{1 + \sin \theta}{\cos^2 \theta}

= \dfrac{1 + \sin \theta}{1 - \sin^2 \theta}

= \dfrac{1 + \sin \theta}{(1 + \sin \theta)(1 - \sin \theta)}

= \dfrac{1}{1 - \sin \theta}

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1.13mins
4 Simplifying Trig Expression 1+sinx:cos^2x

Simplifying Trig Expression

ex Simplify \dfrac{\sin \theta + \sin^2 \theta}{\cos \theta(1 + \sin \theta)}

= \dfrac{\sin \theta(1 + \sin \theta)}{\cos \theta (1 + \sin \theta)}

= \tan \theta

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1.05mins
5 Simplifying Trig Expression sinx +sin^2x:cosx1+sinx

Introduction to Proving Identities

LS = RS

Prove that LS is the same as RS.

ex \dfrac{1}{\cos \theta} + \tan \theta = \dfrac{1 + \sin \theta}{\cos \theta}

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2.36mins
6 Introduction to Proving Identities

Proving Identity ex1

ex 1 - \cos^2 \theta = \sin \theta \cos \theta \tan \theta

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1.40mins
7 Proving Identity ex1

Proving Identity ex2

ex Prove 2 \cos^2 \theta + \sin^2 \theta - 1 = \cos^2 \theta

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1.17mins
8 Proving Identity ex2

Proving Identity ex3

ex Prove \tan \theta + \dfrac{1}{\tan \theta} = \dfrac{1}{\sin \theta \cos \theta}

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0.55mins
9 Proving Identity ex3