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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

advfnsnelsonCh7.4lecture
3 Videos

Prove that `\frac{\sin 3x}{\sin x \cos x} = 4 \cos x - \sec x`

.

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3.41mins

Trig Identity using Double Angles

Prove that

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

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Trig Identity using Tangent Double Angle Example