Purchase this Material for $5

You need to sign up or log in to purchase.

Subscribe for All Access
You need to sign up or log in to purchase.

Solutions
39 Videos

Prove each identity by writing all trigonometric ratios in terms of `x`

, `y`

, and `r`

. State the restrictions on `\theta`

.

```
\displaystyle
\text{cot }\theta = \frac{\text{cos }\theta}{\text{sin }\theta}
```

Buy to View

0.44mins

Q1a

Prove each identity by writing all trigonometric ratios in terms of `x`

, `y`

, and `r`

. State the restrictions on `\theta`

.

```
\displaystyle
\text{tan }\theta \text{ cos }\theta = \text{ sin }\theta
```

Buy to View

0.40mins

Q1b

Prove each identity by writing all trigonometric ratios in terms of `x`

, `y`

, and `r`

. State the restrictions on `\theta`

.

```
\displaystyle
\text{csc }\theta = \frac{1}{\text{sin }\theta}
```

Buy to View

0.41mins

Q1c

`x`

, `y`

, and `r`

. State the restrictions on `\theta`

.

```
\displaystyle
\text{cos }\theta \text{ sec }\theta = 1
```

Buy to View

0.24mins

Q1d

Simplify each expression.

```
\displaystyle
(1 - \text{sin }\alpha)(1 + \text{sin }\alpha)
```

Buy to View

0.42mins

Q2a

Simplify each expression.

```
\displaystyle
\frac{\text{tan }\alpha}{\text{sin }\alpha}
```

Buy to View

0.31mins

Q2b

Simplify each expression.

```
\displaystyle
\text{cos}^{2}\alpha + \text{sin}^{2}\alpha
```

Buy to View

0.07mins

Q2c

Simplify each expression.

```
\displaystyle
\text{cot }\alpha \text{ sin }\alpha
```

Buy to View

0.14mins

Q2d

Factor each expression.

```
\displaystyle
1 - \text{cos}^{2}\theta
```

Buy to View

0.21mins

Q3a

Factor each expression.

```
\displaystyle
\text{sin}^{2}\theta - \text{cos}^{2}\theta
```

Buy to View

0.16mins

Q3b

Factor each expression.

```
\displaystyle
\text{sin}^{2}\theta - 2\text{sin }\theta + 1
```

Buy to View

0.25mins

Q3c

Factor each expression.

```
\displaystyle
\text{cos }\theta - cos^{2}\theta
```

Buy to View

0.17mins

Q3d

Prove that `\frac{\cos^{2}\phi}{1 - \sin\phi} = 1 + \sin\phi`

, where `\sin\phi \neq 1`

, by expressing `\cos^{2}\phi`

in terms of `\sin \phi`

.

Buy to View

1.02mins

Q4

Prove the identity. State any restrictions on the variables.

```
\displaystyle
\frac{\sin x}{\tan x} = \cos x
```

Buy to View

0.30mins

Q5a

Prove the identity. State any restrictions on the variables.

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

Buy to View

1.00mins

Q5b

Prove the identity. State any restrictions on the variables.

```
\displaystyle
\frac{1}{\cos\alpha} + \tan\alpha = \frac{1 + \sin\alpha}{\cos\alpha}
```

Buy to View

0.29mins

Q5c

Prove the identity. State any restrictions on the variables.

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

Buy to View

0.46mins

Q5d

Mark claimed that `\displaystyle \frac{1}{\cot \theta} = \tan \theta`

is an identity. Nancy let `\theta = 30^o`

and found that both sides of the equation worked out to 1 . She said that this proves that the equation is an identity. Is Nancy's reasoning correct? Explain.

Buy to View

1.25mins

Q6

Simplify each trigonometric expression.

```
\displaystyle
\sin \theta \cot \theta - \sin \theta \cos \theta
```

Buy to View

0.34mins

Q7a

Simplify each trigonometric expression.

```
\displaystyle
\cos\theta(1 + \sec\theta)(\cos\theta - 1)
```

Buy to View

0.57mins

Q7b

Simplify each trigonometric expression.

```
\displaystyle
(\sin x +\cos x)(\sin x -\cos x) + 2\cos^2x
```

Buy to View

0.25mins

Q7c

Simplify each trigonometric expression.

```
\displaystyle
\frac{\csc^{2}\theta - 3\csc\theta + 2}{\csc^{2}\theta - 1}
```

Buy to View

1.16mins

Q7d

Prove each identity. State any restrictions on the variables.

`\dfrac{\sin^{2}\phi}{1 - \cos\phi} = 1 + \cos\phi`

Buy to View

0.46mins

Q8a

Prove each identity. State any restrictions on the variables.

`\dfrac{\tan^{2}\alpha}{1 + \tan^{2}\alpha} = \sin^{2}\alpha`

Buy to View

0.52mins

Q8b

Prove each identity. State any restrictions on the variables.

`\cos^{2}x = (1 - \sin x)(1 + \sin x)`

Buy to View

0.21mins

Q8c

Prove each identity. State any restrictions on the variables.

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

Buy to View

0.25mins

Q8d

Prove each identity. State any restrictions on the variables.

`\sin^{4}\alpha - cos^{4}\alpha = \sin^{2}\alpha - \cos^{2}\alpha`

Buy to View

0.57mins

Q8e

Prove each identity. State any restrictions on the variables.

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

Buy to View

0.43mins

Q8f

Is `\csc^{2}\theta + \sec^{2}\theta = 1`

an identity? Prove that it is true or demonstrate why it is false.

Buy to View

0.34mins

Q10

Prove that `\sin^{2}x(1 + \dfrac{1}{\tan^{2}x}) = 1`

, where `\sin x \neq 0`

.

Buy to View

0.42mins

Q11

Prove the identity. State any restrictions on the variables.

```
\displaystyle
\frac{\sin^2x + 2\cos x - 1}{\sin^2 x +3\cos x -3} = \frac{\cos^2x + \cos x}{-\sin^2 x}
```

Buy to View

2.51mins

Q12a

Prove the identity. State any restrictions on the variables.

```
\displaystyle
\sin^2x -\cos^2x -\tan^2x = \frac{2\sin^2x -2\sin^4 x -1}{1- \sin^2 x}
```

Buy to View

3.16mins

Q12b

Show how you can create several new identities from the identity
`\sin^2x + \cos^2 x=1`

by adding, subtracting, multiplying, or dividing
both sides of the equation by the same expression.

Buy to View

1.33mins

Q13

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

`(1 - \cos^{2}x)(1 - \tan^{2}x) = \frac{\sin^{2}x - 2\sin^{4}x} {1 - \sin^{2}x}`

Buy to View

4.56mins

Q14i

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

`1 - 2\cos^{2}\phi = \sin^{4}\phi - \cos^{4}\phi`

Buy to View

1.05mins

Q14ii

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

```
\displaystyle
\dfrac{\sin x\tan x}{\sin x + \tan x} = \sin x \tan x
```

Buy to View

0.55mins

Q14iii

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

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

Buy to View

1.02mins

Q14iv

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

```
\displaystyle
\frac{1 -\cos x}{\sin x} = \frac{\sin x}{1 + \cos x}
```

Buy to View

1.04mins

Q14v

- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.

```
\displaystyle
\frac{\sin x}{ 1 + \cos x} = \csc x - \cot x
```

Buy to View

1.12mins

Q14vi

Lecture on Identity
9 Videos

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`

Buy to View

7.22mins

1 Introduction to Trig Ratios and to Identity

`\cot x \sin x`

*ex* Simplify `cot \theta \times \sin \theta`

.

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

`= \cos \theta`

Buy to View

0.33mins

2 Simplifying cotx sin x

`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`

Buy to View

1.14mins

3 Simplifying Trig Expression 1+tan^2x

`\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}`

Buy to View

1.13mins

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

*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`

Buy to View

1.05mins

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

LS = RS

Prove that LS is the same as RS.

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

Buy to View

2.36mins

6 Introduction to Proving Identities

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

Buy to View

1.40mins

7 Proving Identity ex1

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

Buy to View

1.17mins

8 Proving Identity ex2

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

Buy to View

0.55mins

9 Proving Identity ex3