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Solutions
39 Videos

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

, `y`

, and `r`

. State the restrictions on `\theta`

.

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

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0.44mins

Q1a

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

, `y`

, and `r`

. State the restrictions on `\theta`

.

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

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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}
```

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0.41mins

Q1c

`x`

, `y`

, and `r`

. State the restrictions on `\theta`

.

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

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0.24mins

Q1d

Simplify each expression.

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

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0.42mins

Q2a

Simplify each expression.

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

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0.31mins

Q2b

Simplify each expression.

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

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0.07mins

Q2c

Simplify each expression.

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

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0.14mins

Q2d

Factor each expression.

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

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0.21mins

Q3a

Factor each expression.

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

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0.16mins

Q3b

Factor each expression.

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

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0.25mins

Q3c

Factor each expression.

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

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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`

.

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1.02mins

Q4

Prove the identity. State any restrictions on the variables.

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

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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}
```

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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}
```

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0.29mins

Q5c

Prove the identity. State any restrictions on the variables.

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

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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.

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1.25mins

Q6

Simplify each trigonometric expression.

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

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0.34mins

Q7a

Simplify each trigonometric expression.

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

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0.57mins

Q7b

Simplify each trigonometric expression.

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

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0.25mins

Q7c

Simplify each trigonometric expression.

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

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Q7d

Prove each identity. State any restrictions on the variables.

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

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0.46mins

Q8a

Prove each identity. State any restrictions on the variables.

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

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0.52mins

Q8b

Prove each identity. State any restrictions on the variables.

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

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0.21mins

Q8c

Prove each identity. State any restrictions on the variables.

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

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0.25mins

Q8d

Prove each identity. State any restrictions on the variables.

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

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0.57mins

Q8e

Prove each identity. State any restrictions on the variables.

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

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0.43mins

Q8f

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

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

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0.34mins

Q10

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

, where `\sin x \neq 0`

.

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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}
```

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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}
```

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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.

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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}`

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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`

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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
```

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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
```

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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}
```

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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
```

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Q14vi

Lecture on Identity
10 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`

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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`

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0.33mins

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`

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1.14mins

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}`

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1.13mins

Simplifying Trig Expression 1+sinx:cos^2x

`\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} `

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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}`

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

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Introduction to Proving Identities

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Proving Identity ex1

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

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Proving Identity ex2

*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} `

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Proving Identity ex3