4.6 Trig Identities
Chapter
Chapter 4
Section
4.6
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

7.22mins
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

0.33mins
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

1.14mins
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}

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

# Simplifying Trig Expression

\displaystyle \begin{aligned} & \tan ^{2} \theta-\sec ^{2} \theta \=& \frac{\sin ^{2} \theta}{\cos ^{2} \theta}-\frac{1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\sin ^{2} \theta-\cos ^{2} \theta}{\cos ^{2} \theta}=-1 \end{aligned}

1.05mins
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}

\displaystyle \begin{aligned} L S &=\frac{1}{\cos \alpha}+\frac{\sin \alpha}{\cos \alpha} \\ &=\frac{1+\sin \alpha}{\cos \alpha}=R .5 . \end{aligned}

2.36mins
Introduction to Proving Identities

## Proving Identity with Relative Angles

1.40mins
Proving Identity ex1

## Proving Identity ex2

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

1.17mins
Proving Identity ex2

## Proving Identity ex3

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

\displaystyle \begin{aligned} & \tan \theta+\frac{1}{\tan \theta}=\frac{1}{\sin \theta \cos \theta} \\ L S &=\frac{\sin \theta}{\cos \theta} s+\frac{\cos \theta}{\sin \theta} \frac{c}{c} \\ &=\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta} \\ &=\frac{1}{\sin \theta \cos \theta}=R S \end{aligned}

0.55mins
Proving Identity ex3
Solutions 50 Videos

Using Pythagorean identities, state an equivalent expression.

 \displaystyle \sin^2 \theta 

Q1a

Using Pythagorean identities, state an equivalent expression.

 \displaystyle \tan^2 \theta 

Q1b

Using Pythagorean identities, state an equivalent expression.

 \displaystyle \sec^2 \theta 

Q1c

Using Pythagorean identities, state an equivalent expression.

 \displaystyle 1 - \sin^2 \theta 

Q1d

Using Pythagorean identities, state an equivalent expression.

 \displaystyle 1 - \csc^2 \theta 

Q1e

Using Pythagorean identities, state an equivalent expression.

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

Q1f

Using Pythagorean identities, state an equivalent expression.

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

Q1g

Express each expression in a simpler form.

 \displaystyle \cos \theta \sec \theta 

Q2a

Express each expression in a simpler form.

 \displaystyle \tan \theta \cos \theta 

Q2b

Express each expression in a simpler form.

 \displaystyle \tan \theta + \cot \theta 

Q2c

Express each expression in a simpler form.

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

Q2d

Express each expression in a simpler form.

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

Q2e

Express each expression in a simpler form.

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

Q2f

Factor to simplify each expression.

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

Q3a

Factor to simplify each expression.

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

Q3b

Factor to simplify each expression.

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

Q3c

Factor to simplify each expression.

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

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 

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 

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 

Q4c

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

Q5

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

Q7

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

Q8

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

Q9

Prove each identity.

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

Q11a

Prove each identity.

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

Q11b

Prove each identity.

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

Q11c

Prove each identity.

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

Q11d

Simplify

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

Q12

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

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.

Q15

Prove that

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

Q16

Prove that

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

Q17

Prove that

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

Q18

Prove that

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

Q19

Prove that

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

Q20

Prove that

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

Q21

Prove that

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

Q22

Prove that

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

Q23

Prove that

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

Q24

Prove that

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

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 

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 

Q26b

Prove that

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

Q28

Prove that

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

Q29

Prove that

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

Q30

Prove that

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

Q31

Prove that

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

\displaystyle \frac{1 -\tan^2\theta}{\tan \theta -\tan^2\theta} = 1 + \frac{1}{\tan \theta} 
Prove that \tan^2\theta(1 + \cot^2\theta) = \sec^2\theta