Free Printable Trigonometric Identities Challenge

1.

Simplify the expression

a)

sin²θ

b)

cosθ

c)

tanθ

d)

1- sin²θ

2.

Simplify the expression

a)

-1

b)

sin θ

c)

csc θ

d)

1

3.

Simplify the expression

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

4.

Simplify the expression

a)

sinθ

b)

cot²θ

c)

tan²θ

d)

cos²θ

5.

cotθ=

a)

cosθ/sinθ

b)

sinθ/cosθ

c)

tanθ

d)

1/cosθ

6.

sin²θ=

a)

1 - cos²θ

b)

1 - sin²θ

c)

sec²θ - 1

d)

csc²θ - 1

7.

*

a)

csc² x

b)

sec² x

c)

cot² x

d)

tan² x

8.

a)

csc x

b)

sec x

c)

1/sec x

d)

cos x

9.

a)

sinθ

b)

cos²θ

c)

sin²θ

d)

1- sin²θ

10.

Choose all valid strategies for proving an identity

a)

Choose the most complicated looking side to work with

b)

Convert everything in terms of sine and cosine

c)

multiply everything by $\pi$

d)

Combine or Separate Fractions

e)

Multiplying by the Conjugate

11.

Which of the following is NOT an Identity under Reciprocal Identity?

a)

$\csc\theta\ =\ \frac{1}{\sin\theta}$

b)

$\sec\theta\ =\ \frac{1}{\cos\theta}$

c)

$\tan\theta\ =\ \frac{\sin\theta}{\cos\theta}$

d)

$1\ =\ \cot\theta\tan\theta$

12.

Simplify\ \frac{1\ +\ \tan^2\theta}{\csc^2\theta}

a)

$\cos^2\theta$

b)

$\sin^2\theta$

c)

$\tan^2\theta$

d)

$1$

13.

$Simplify\ \frac{\sin^2x}{\sec^2-1}$

a)

$\sin^2x$

b)

$\cos^2x$

c)

0

d)

1

14.

Which of the following should be written in Item number 1.

a)

1

b)

$\frac{\sin x}{\sin x}$

c)

$\frac{\cos x}{\cos x}$

d)

$\frac{\cos x}{\sin x}$

15.

Which of the following should be written in Item number 2.

a)

cos x - 1

b)

1- cos x

c)

sinx - cos x

d)

1 - sin x

16.

Which of the following should be written in Item number 3.

a)

1 - cosx

b)

$\frac{\cos x}{\cos x}$

c)

$1\ -\ \cos^2x$

d)

1

17.

Which of the following should be written in Item number 4.

a)

$1-\cos x$

b)

$1\ +\ \cos x$

c)

$1\ -\ \cos^2x$

d)

$1\ +\ \cos^2x$

18.

Which of the following should be written in Item number 1.

a)

1

b)

$\sin x\ +\ \cos x$

c)

$\sin^2x\ +\ \cos^2x$

d)

$\sin^2x\ -\ \cos^2x$

19.

Which of the following should be written in Item number 2.

a)

$\cos^2x$

b)

$\frac{1}{\cos x}$

c)

$\csc^2x$

d)

$\frac{1}{\cos^2x}$

20.

Which of the following should be written in Item number 3.

a)

$\csc x$

b)

$\csc^2x$

c)

$\frac{1}{\cos^2x}$

d)

$\sin^2x$

21.

$\tan\theta\ =$

a)

$\frac{\sin\theta}{\cos\theta}$

b)

$\frac{\cos\theta}{\sin\theta}$

c)

$\frac{1}{\cos\theta}$

d)

$\cot\theta$

22.

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

Solve for $\cos^2\theta$ .

a)

$\cot^2\theta+1$

b)

$\sec^2\theta-1$

c)

$\tan\theta+1$

d)

$1-\sin^2\theta$

23.

$\tan\theta\cos\theta$ can be written in a single trigonometric identity as:

a)

$\cos\theta$

b)

$\sin\theta$

c)

$\sec\theta$

d)

$\cot\theta$

24.

$\frac{1-\cos^2\theta}{\cos^2\theta}$ can be written in a single trigonometric identity as:

a)

$\cos^2\theta$

b)

$\sin^2\theta$

c)

$\sec^2\theta$

d)

$\tan^2\theta$

25.

$\cos\theta\ \csc\theta$ can be written in a single trigonometric identity as:

a)

$\cos\theta$

b)

$\sin\theta$

c)

$\sec\theta$

d)

$\cot\theta$

26.

Simplify

$\cot^2\theta\left(1+\tan^2\theta\right)$

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

27.

Simplify

$\sin\theta\left(\csc\theta-\sin\theta\right)$

a)

$\sec\theta$

b)

$\cos^2\theta$

c)

$\sin^2\theta$

d)

$\frac{\sin^2\theta}{\cos^2\theta}$

28.

Simplify

$\tan x\csc x\cos x$

a)

$\frac{1}{\cos x}$

b)

1

c)

$\cot x$

d)

-1

29.

Simplify $\left(\sec\theta-1\right)\left(\sec\theta+1\right)$

a)

$2\sec\theta$

b)

$\cot^2\theta$

c)

$\tan^2\theta$

d)

$\sec^2\theta+1$

30.

Simplify

$\csc\ x\left(\cos x+\sin x\right)$

a)

csc x

b)

tan x + 1

c)

cot x

d)

cot x + 1

31.

Simplify $\tan x\left(\cot x\ +\ \csc x\right)$

a)

$1\ +\ \sec x$

b)

$1\ +\ \csc x$

c)

$\sec x-1$

d)

$\tan^2x$

32.

$\frac{1-\cos^2\theta}{\tan^2\theta}$ can be simplified as

a)

$\cot^2\theta$

b)

$\tan^2\theta$

c)

$\sin^2\theta$

d)

$\cos^2\theta$

33.

What happens when you multiply two reciprocal functions?

a)

You get a pythagorean identity

b)

It equals 1

c)

You get a quotient identity

d)

It equals 0

34.

Which of the following would be a step to prove the following identity? $\csc x-\sec x=\frac{\cos x-\sin x}{\sin x\cos x}$

a)

$\frac{1}{\sin x}-\frac{1}{\cos x}$

b)

$\frac{\cos x-\sin x}{\sin x}$

c)

$\frac{\cos x-\sin x}{\cos x}$

d)

$\frac{1}{\sin x\cos x}$

35.

One step in proving the identity below would be:

\frac{\sec^2\theta-1}{\sec^2\theta}

a)

$\frac{\sec^2\theta}{\sec^2\theta}-\frac{1}{\sec^2\theta}$

b)

$\frac{\tan^2\theta}{\sec^2\theta}$

c)

Both A and B

d)

Neither A nor B

36.

$\frac{\tan\theta+\cot\theta}{\tan\theta}$ can be simplified as:

a)

$\csc^2\theta$

b)

$\cot^2\theta$

c)

$\sin^2\theta$

d)

$\cos^2\theta$

37.

$\sin\theta\cot\theta\ \sec\theta$

a)

$0$

b)

$1$

c)

$\sin\theta$

d)

$\cos\theta$

38.

Simplify $\left(\sec\theta-1\right)\left(\sec\theta+1\right)$

a)

2secθ

b)

cot²θ

c)

tan²θ

d)

sec²θ + 1

39.

Simplify

$\csc\ x\left(\cos x+\sin x\right)$

a)

csc x

b)

tan x + 1

c)

cot x

d)

cot x + 1

40.

Rewrite $\tan x$ in terms of $\sin x$ and $\cos x$

a)

$\tan x=\frac{\cos x}{\sin x}$

b)

$\tan x=\frac{1}{\cot x}$

c)

$\tan x=\frac{\sin x}{\cos x}$

d)

$\tan x=\frac{opposite}{adjacent}$

41.

$\tan\theta\cos\theta$ can be written in a single trigonometric identity as:

a)

$\cos\theta$

b)

$\sin\theta$

c)

$\sec\theta$

d)

$\cot\theta$

42.

$\frac{1-\cos^2\theta}{\cos^2\theta}$ can be written in a single trigonometric identity as:

a)

$\cos^2\theta$

b)

$\sin^2\theta$

c)

$\sec^2\theta$

d)

$\tan^2\theta$

43.

$\cos\theta\ \csc\theta$ can be written in a single trigonometric identity as:

a)

$\cos\theta$

b)

$\sin\theta$

c)

$\sec\theta$

d)

$\cot\theta$

44.

Simplify

$\cot^2\theta\left(1+\tan^2\theta\right)$

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

45.

Simplify

$\sin\theta\left(\csc\theta-\sin\theta\right)$

a)

$\sec\theta$

b)

$\cos^2\theta$

c)

$\sin^2\theta$

d)

$\frac{\sin^2\theta}{\cos^2\theta}$

46.

Simplify

$\tan x\csc x\cos x$

a)

$\frac{1}{\cos x}$

b)

1

c)

$\cot x$

d)

-1

47.

$\frac{1-\cos^2\theta}{\tan^2\theta}$ can be simplified as

a)

$\cot^2\theta$

b)

$\tan^2\theta$

c)

$\sin^2\theta$

d)

$\cos^2\theta$

48.

Which of the following would be a step to prove the following identity? $\csc x-\sec x=\frac{\cos x-\sin x}{\sin x\cos x}$

a)

$\frac{1}{\sin x}-\frac{1}{\cos x}$

b)

$\frac{\cos x-\sin x}{\sin x}$

c)

$\frac{\cos x-\sin x}{\cos x}$

d)

$\frac{1}{\sin x\cos x}$

49.

One step in proving the identity below would be:

\frac{\sec^2\theta-1}{\sec^2\theta}

a)

$\frac{\sec^2\theta}{\sec^2\theta}-\frac{1}{\sec^2\theta}$

b)

$\frac{\tan^2\theta}{\sec^2\theta}$

c)

Both A and B

d)

Neither A nor B

50.

$\frac{\tan\theta+\cot\theta}{\tan\theta}$ can be simplified as:

a)

$\csc^2\theta$

b)

$\cot^2\theta$

c)

$\sin^2\theta$

d)

$\cos^2\theta$

51.

$\sin\theta\cot\theta\ \sec\theta$

a)

$0$

b)

$1$

c)

$\sin\theta$

d)

$\cos\theta$

52.

Simplify

$\csc\ x\left(\cos x+\sin x\right)$

a)

csc x

b)

tan x + 1

c)

cot x

d)

cot x + 1

53.

Simplify

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

54.

Simplify.

a)

sinθ

b)

cos²θ

c)

sin²θ

d)

1- sin²θ

55.

Simplify.

a)

sin x

b)

cos x

c)

tan x

d)

1

56.

5) tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

57.

a)

cos²x

b)

sin²x

c)

cot²x

d)

tan²x

58.

Simplify: $\frac{\sin^2x+\cos^2x+\cot^2x}{1+\tan^2x}$

a)

$\cot^2x$

b)

$\tan^2x$

c)

$\csc^2x$

d)

$\sec^2x$

59.

Simplify the trig expression.

a)

sinx

b)

-sinx

c)

cosx

d)

2sinx

60.

Simplify: $\frac{\cos^2\left(-x\right)\csc x}{\cot x}$

a)

$-\sin x$

b)

$-\cos x$

c)

$\sin x$

d)

$\cos x$

61.

Given the Csc(x) = 1, then the Sin(x) = ?

a)

1

b)

-1

c)

0

d)

Undefined

62.

Given the Cos(x) = -2/3, then the Sec(x) = ?

a)

-2/3

b)

-3/2

c)

3/2

d)

2/3

63.

Which of these is true because of even symmetry?

a)

$\cos\left(-40\degree\right)=\cos\left(40\degree\right)$

b)

$\sin\left(-56\degree\right)=-\sin\left(56\degree\right)$

c)

$\tan\left(44\degree\right)=\cot\left(46\degree\right)$

d)

$\tan\left(12\degree\right)=\frac{1}{\cot\left(78\degree\right)}$

64.

$\cos\left(-\frac{4\pi}{5}\right)=$
Complete this equation using even symmetry.

a)

$\frac{1}{\sec\left(-\frac{4\pi}{5}\right)}$

b)

$\sin\left(-\frac{4\pi}{5}\right)$

c)

$-\cos\left(\frac{4\pi}{5}\right)$

d)

$\cos\left(\frac{4\pi}{5}\right)$

Trigonometric Identities Challenge

Total questions: 64

Worksheet time: 3hrs 21mins

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