Deck 32: Verifying Trigonometric Equations

Question 1

Which of the following expressions is equivalent to $\frac { 5 } { \tan \beta } + 5 \tan \beta$ ? (? ? ?/2+?n,where n is a whole number) ?

A) $\frac { 5 \cos ^ { 2 } \beta } { \cot \beta }$
B) $\frac { 5 \tan ^ { 2 } \beta } { \sec \beta }$
C) $\frac { 5 \sec ^ { 2 } \beta } { \tan \beta }$
D) $\frac { 5 \sec ^ { 2 } \beta } { \cot \beta }$
E) $\frac { 5 \cos ^ { 2 } \beta } { \tan \beta }$

Answer 1

$\frac { 5 \sec ^ { 2 } \beta } { \tan \beta }$

Question 2

Evaluate the following expression.(a ? ?/2+?n,where n is a whole number)? $$\frac { 5 \tan ^ { 2 } \theta } { \sec \theta }$$ ?

A)5 cos ? tan ?
B)5 tan ? sec ?
C)5 sin ? tan ?
D)5 cos ? cot ?
E)5 cot ? sec ?

Answer 2

5 sin ? tan ?

Question 3

Evaluate the following expression. ​
(sin²x - sin⁴x)cos x
​

A)cos⁴x sin²x
B)cos²x sin²x
C)cos²x sin⁶x
D)cos³x sin²x
E)cos x sin x

Answer 3

cos³x sin²x

Question 4

Which of the following expressions is equivalent to $$\frac { 5 \cot ^ { 2 } t } { \csc t }$$ ? (t ? ?n,where n is a whole number) ?

A)? $\frac { 5 - 5 \sin ^ { 2 } t } { \sin t }$
B) $\frac { - 1 - \sin ^ { 2 } t } { 5 \sin t }$
C) $\frac { 5 - 5 \cos ^ { 2 } t } { \cos t }$
D) $\frac { 5 + 5 \sin ^ { 2 } t } { \sin t }$
E) $\frac { 5 + 5 \cos ^ { 2 } t } { \cos t }$

Question 5

Evaluate the following expression. ​
Cos² β - sin² β
​

A)1 + 2 sin² β
B)1 + 2 cos² β
C)2 cos² β - 1
D)2 sin² β - 1
E)1 - 2 cos² β

Question 6

Which of the following expressions is equivalent to 2 - 4 cos²x + 4 cos⁴x? ​

A)2 sin²x + 2 cos²x
B)2 sin⁴x - 2 cos²x
C)2 sin2x + 2 cos⁴x
D)2 sin⁴x + 2 cos⁴x
E)2 sin⁴x - 2 cos⁴x

Question 7

Use the cofunction identities to evaluate the expression without using a calculator. ​
Tan²73° + cot²14° - sec²76° - csc²17°
​

A)3
B)1
C)-1
D)-2
E)2

Question 8

Use the cofunction identities to evaluate the expression without using a calculator. ​
Sin²25° + sin²65°
​

A)-1
B)-2
C)1
D)2
E)3

Question 9

Which of the following expressions is equivalent to $$\frac { 2 \cot ^ { 3 } t } { \csc t }$$ ? (a ? ?n,where n is a whole number) ?

A)sin t (2 - 2 csc²t)
B)cos t (2 csc²t - 2)
C)cos t (2 - 2 csc²t)
D)sin t (2 csc²t - 2)
E)tan t (2 csc²t - 2)

Question 10

Evaluate the following expression.(x ≠ π/2+πn,where n is a whole number) ​
Tan⁷x sec²x - tan⁷x
​

A)9 tan⁹x
B)tan²x
C)tan⁹x
D)7 tan⁷x
E)tan⁷x

Question 11

Which of the following expressions is equivalent to 3 sin² α - 3 sin⁴ α? ​

A)3 cos⁴ α + cos² α
B)-cos² α - 3 cos⁴ α)
C)3 (cos² α + cos⁴ α)
D)3 (cos² α - cos⁴ α)
E)-3 (cos² α - cos⁴ α)

Question 12

Evaluate the following expression.(x ≠ π/2+πn,where n is a whole number) ​
Sec⁵x (sec x tan x)- ​sec³x (sec x tan x)
​

A)​sec⁴x tan x
B)​4 sec x tan⁴ x
C)​sec³x tan⁵x
D)​sec⁴x tan³ x
E)​sec³x tan³x

Question 13

Evaluate the following expression. ​
Cot²t (2 sec²t - 2)
​

A)2
B)0
C)3
D)4
E)1

Question 14

Evaluate the following expression.(x ≠ π/2+πn,where n is a whole number) ​
(tan⁵x + tan⁷x)sec²x
​

A)sec4x tan¹²x
B)sec4x tan⁵x
C)7 sec4x tan⁵x
D)5 sec4x tan⁵x
E)sec12x tan⁵x

Question 15

Evaluate the following expression.(x ≠ π/2+πn,where n is a whole number) ​
2 cos x + 2 sin x tan x
​

A)-2 sec x
B)2 sec x
C)4 cos x
D)-4 cos x
E)3 sin x

Question 16

Evaluate the following expression. ​
2(1 + sin α)(1 - sin α)
​

A)-3 sin² α
B)3 cos² α
C)-2 cos² α
D)2 sin² α
E)2 cos² α

Question 17

Evaluate the following expression. ​
4 tan t cot t
​

A)3
B)5
C)4
D)2
E)6

Question 18

Evaluate the following expression. ​
4 sec y cos y
​

A)5
B)3
C)6
D)7
E)4

Question 19

Use the cofunction identities to evaluate the expression without using a calculator. ​
Cos²35° + cos²50° + cos²40° + cos²55°
​

A)-2
B)1
C)-1
D)3
E)2

Question 20

Use the cofunction identities to evaluate the expression without using a calculator. ​
Cos²65° + cos²25°
​

A)3
B)4
C)-3
D)-1
E)1

Question 21

Evaluate the following expression.(? ? ?n,where n is a whole number)? $\frac { 5 \cos 4 \theta \cot 4 \theta } { 1 - \sin 4 \theta } - 5$ ?

A)5 sec 4?
B)- 5 csc 4?
C)5 csc 4?
D)- 5 sec 4?
E)5 cot 4?

Question 22

Which of the following expressions is equivalent to $\frac { \cot 3 x } { \sec 3 x }$ ? (x ? (?/3)n,where n is a whole number) ?

A)-3 sec x - sin x
B)csc 3x - sin 3x
C)csc 3x + sin 3x
D)-3 csc x + sin x
E)- csc x -3 sin x

Question 23

Evaluate the following expression.? $$2 \cos ^ { 2 } \beta + 2 \cos ^ { 2 } \left( \frac { \pi } { 2 } - \beta \right)$$ ?

A)3
B)- 2
C)2
D)-3
E)0

Question 24

Evaluate the following expression.(x ? ?/10 + ?n,where n is a whole number)? $5 \cos 5 x - \frac { 5 \cos 5 x } { 1 - \tan 5 x }$ ?

A) $\frac { 10 \sin 5 x \cos 5 x } { \sin 5 x - \cos 5 x }$
B) $\frac { - 5 \sin 5 x \cos 5 x } { \sin 5 x + \cos 5 x }$
C) $\frac { 5 \sin 5 x \cos 5 x } { \sin 5 x - \cos 5 x }$
D) $\frac { - 5 \sin 5 x \cos 5 x } { \sin 5 x - \cos 5 x }$
E) $\frac { 5 \sin 5 x \cos 5 x } { \sin 5 x + \cos 5 x }$

Question 25

Evaluate the following expression.(x ? (?/8)n,where n is a whole number)? $\frac { 3 } { \tan 4 x } + \frac { 3 } { \cot 4 x }$ ?

A)3 (tan 4x - cot 4x)
B)-4 (tan 4x - cot 4x)
C)3 (tan 4x + cot 4x)
D)- (tan 4x - cot 4x)
E)- (tan 4x + cot 4x)

Question 26

Evaluate the following expression.(? ? ?/2 + ?n,where n is a whole number)? $\frac { 3 + 3 \sin \theta } { \cos \theta } + \frac { 3 \cos \theta } { 1 + \sin \theta }$ ?

A)6 sec ?
B)sec 6?
C)- sec ?
D)sec ?
E)-6 sec ?

Question 27

Which of the following expressions is equivalent to $$\frac { 3 \tan 8 x + 3 \tan 8 y } { 1 - \tan 8 x \tan 8 y }$$ ? (x ? ?n,y ? ?n,where n is a whole number) ?

A) $\frac { 8 \cot 3 x + 8 \cot 3 y } { \cot 3 x \cot 3 y - 1 }$
B) $\frac { 3 \cot 8 x + 3 \cot 8 y } { \cot 8 x \cot 8 y + 1 }$
C) $$\frac { 3 \cot 8 x + 3 \cot 8 y } { \cot 8 x \cot 8 y - 1 }$$
D) $\frac { 3 \cot 8 x - 3 \cot 8 y } { \cot 8 x \cot 8 y - 1 }$
E) $\frac { 8 \cot 3 x + 8 \cot 3 y } { \cot 3 x \cot 3 y + 1 }$

Question 28

Which of the following expressions is equivalent to the given expression $$\sqrt { \frac { 25 - 25 \cos \theta } { 1 + \cos \theta } }$$ ? (? ? ?n,where n is a whole number) ?

A) $\frac { 1 - \cos \theta } { | \sin \theta | }$
B) $\frac { 5 - 5 \cos \theta } { | \sin \theta | }$
C) $\frac { 5 + 5 \sin \theta } { | \cos \theta | }$
D) $\frac { 5 - 5 \sin \theta } { | \cos \theta | }$
E) $\frac { 5 + 5 \cos \theta } { | \sin \theta | }$

Question 29

Which of the following expressions is equivalent to the given expression $\sqrt { \frac { 9 + 9 \sin \theta } { 1 - \sin \theta } }$ ? (? ? ?/2 + ?n,where n is a whole number) ?

A) $\frac { 1 + \sin \theta } { | \cos \theta | }$
B) $\frac { 3 - 3 \cos \theta } { | \sin \theta | }$
C) $\frac { 3 - 3 \sin \theta } { | \cos \theta | }$
D) $\frac { 3 + 3 \cos \theta } { | \sin \theta | }$
E) $\frac { 3 + 3 \sin \theta } { | \cos \theta | }$

Question 30

Evaluate the following expression.(x ? ?n,where n is a whole number)? $\frac { 3 } { \sin x } - \frac { 3 } { \csc x }$ ?

A)3 (csc x - sin x)
B)(csc x + sin x)
C)- 3 (csc x - sin x)
D)3 (csc x + sin x)
E)(csc x - sin x)

Question 31

Evaluate the following expression.(x ? ?n,y ? ?/12 + ?n,where n is a whole number)? $\frac { 7 \tan 6 x + 7 \cot 6 y } { \tan 6 x \cot 6 y }$ ?

A)7 tan 6x + 7 cot 6y
B)7 tan 6y - 7 cot 6x
C)7 tan 6y + 7 cot 6x
D)- (7 tan 6y + 7 cot 6x)
E)7 tan 6x - 7 cot 6y

Question 32

Evaluate the following expression.(? ? (?+2?n)/6,where n is a whole number)? $\frac { \sec 3 \theta - 1 } { 1 - \cos 3 \theta }$ ?

A)- cos 3?
B)cos 3?
C)sec 3?
D)- sec 3?
E)sec 3?? - cos 3?

Question 33

Evaluate the following expression.? $$\frac { \cos 2 x - \cos 2 y } { \sin 2 x + \sin 2 y } + \frac { \sin 2 x - \sin 2 y } { \cos 2 x + \cos 2 y }$$ ?

A)1
B)2
C)-2
D)0
E)- 1

Question 34

Evaluate the following expression.(x ≠ (π + 2πn)/10,where n is a whole number) ​
Sec 5x - cos 5x
​

A)- 5 sin 5x cot 5x
B)- sin 5x tan 5x
C)sin 5x csc 5x
D)5 sin 5x cot x
E)sin 5x tan 5x

Question 35

Evaluate the following expression.? $$3 \sec ^ { 2 } y - 3 \cot ^ { 2 } \left( \frac { \pi } { 2 } - y \right)$$ ?

A)0
B)- 3
C)4
D)- 4
E)3

Question 36

Evaluate the following expression.(x ? ?/2 + ?n,where n is a whole number)? $\frac { 11 \cos \left[ \frac { \pi } { 2 } - x \right] } { \sin \left[ \frac { \pi } { 2 } - x \right] }$ ?

A)- 11 tan x
B)11 cot x
C)12 tan x
D)- 11 cot x
E)11 tan x

Question 37

Evaluate the following expression.(x ≠ (π/5)n,where n is a whole number) ​
Csc 5x - sin 5x
​

A)cos 5x cot 5x
B)- 5 sin 5x cot 5x
C)- 5 cos 5x cot 5x
D)sin 5x csc 5x
E)5 sin x cot x

Question 38

Evaluate the following expression.(x ? ?/2 + ?n,where n is a whole number)? $\frac{6 \tan x \cot x}{\cos x}$ ?

A)- 6 tan x
B)6 sec x
C)6 tan x
D)- 6 sec x
E)6 cot x

Question 39

Evaluate the following expression. ?? $5 \tan \left( \frac { \pi } { 2 } - \theta \right) \tan \theta$ ?

A)-6
B)6
C)4
D)5
E)-5

Question 40

Evaluate the following expression. ​
(1 + sin 5y)(1 + sin (- 5y))
​

A)sin²5y
B)cos²2y
C)- sin²5y
D)- cos²5y
E)cos²5y

Question 41

Use the cofunction identities to evaluate the expression below without using calculator. sin²20° + sin²27° + sin²70° + sin²63°

A)1
B) $\frac { 1 } { 2 }$
C)-1
D)?0
E)2

Question 42

Evaluate the following expression.(t ? ?/2+?n,where n is a whole number)? $$2 \sin t \csc \left( \frac { \pi } { 2 } - t \right)$$ ?

A)- 2 tan t
B)2 tan t
C)2 sin t
D)- 2 cot t
E)2 cot t

Question 43

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $$\cos ^ { 2 } 20 ^ { \circ } + \cos ^ { 2 } 57 ^ { \circ } + \cos ^ { 2 } 70 ^ { \circ } + \cos ^ { 2 } 33 ^ { \circ }$$

A)1
B)2
C)-1
D)0
E) $\frac { 1 } { 2 }$

Question 44

Evaluate the expression.? $$\frac { 1 } { \tan x + \cot x }$$ ?

A) $- \sin ^ { 2 } x$
B) $\sin x$
C) $\sin x \cos x$
D) $\cos ^ { 2 } x$
E)0

Question 45

Evaluate the following expression. ?
4 tan x (arcsin x)
?

A) $\frac { - 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
B) $\frac { 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
C) $\frac { - 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$
D) $\frac { 4 x \sqrt { x ^ { 2 } - 1 } } { x ^ { 2 } - 1 }$
E) $\frac { 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$

Question 46

Evaluate the following expression.(x ? ?n,where n is a whole number)? $$5 \sec ^ { 2 } \left( \frac { \pi } { 2 } - x \right) - 5$$ ?

A)5 sec²x
B)- 5 cot²x
C)5 tan²x
D)- 5 tan²x
E)5 cot²x