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Worksheet: Trigonometric Identities

Q1:

Simplify ( 1 βˆ’ πœƒ ) + ( 1 + πœƒ ) t a n t a n 2 2 .

  • A s e c 2 πœƒ
  • B 2 πœƒ c s c 2
  • C c s c 2 πœƒ
  • D 2 πœƒ s e c 2

Q2:

Simplify s i n c o s c s c c o t 2 2 2 2 πœƒ + πœƒ πœƒ βˆ’ πœƒ .

  • A c o s 2 πœƒ
  • B βˆ’ 1
  • C βˆ’ πœƒ c o s 2
  • D1

Q3:

Simplify 1 βˆ’ πœƒ πœƒ βˆ’ πœƒ s i n c s c c o t 2 2 2 .

  • A s i n 2 πœƒ
  • B βˆ’ πœƒ c o s 2
  • C βˆ’ πœƒ s i n 2
  • D c o s 2 πœƒ

Q4:

Simplify s e c s e c t a n 2 2 2 πœƒ βˆ’ 1 πœƒ βˆ’ πœƒ .

  • A s i n 2 πœƒ
  • B βˆ’ πœƒ t a n 2
  • C βˆ’ πœƒ s i n 2
  • D t a n 2 πœƒ

Q5:

Find the value of t a n c o t ( πœ‹ + 𝐴 ) βˆ’ ο€» 𝐴 βˆ’ πœ‹ 2  given 2 1 𝐴 = βˆ’ 2 9 c s c where 3 πœ‹ 2 < 𝐴 < 2 πœ‹ .

  • A 4 1 4 2 0
  • B 2 1 1 0
  • C βˆ’ 4 1 4 2 0
  • D βˆ’ 2 1 1 0

Q6:

Find the possible values of t a n c o t 2 2 πœƒ βˆ’ πœƒ given that t a n c o t πœƒ + πœƒ = 2 4 .

  • A 2 √ 1 4 5 , βˆ’ 2 √ 1 4 5
  • B √ 5 7 4 , βˆ’ √ 5 7 4
  • C 1 7 √ 2 , βˆ’ 1 7 √ 2
  • D 4 8 √ 1 4 3 , βˆ’ 4 8 √ 1 4 3

Q7:

Knowing that 5 + 4 π‘₯ = βˆ’ 1 2 π‘₯ c o s t a n 2 , find t a n π‘₯ .

  • A t a n π‘₯ = βˆ’ 1 2
  • B t a n π‘₯ = 3 2
  • C t a n π‘₯ = βˆ’ 5 2
  • D t a n π‘₯ = βˆ’ 3 2
  • E t a n π‘₯ = 5 2

Q8:

Find the set of values satisfying s e c t a n t a n 2 2 πœƒ βˆ’ πœƒ + √ 3 πœƒ = 0 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

  • A { 1 5 0 , 2 1 0 } ∘ ∘
  • B { 3 0 , 2 1 0 } ∘ ∘
  • C { 2 1 0 , 3 3 0 } ∘ ∘
  • D { 1 5 0 , 3 3 0 } ∘ ∘

Q9:

Find the set of values satisfying s e c t a n t a n 2 2 πœƒ βˆ’ πœƒ βˆ’ πœƒ = 0 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

  • A { 4 5 , 3 1 5 } ∘ ∘
  • B { 1 3 5 , 3 1 5 } ∘ ∘
  • C { 1 3 5 , 2 2 5 } ∘ ∘
  • D { 4 5 , 2 2 5 } ∘ ∘

Q10:

Find the set of values satisfying s e c t a n t a n 2 2 πœƒ βˆ’ πœƒ βˆ’ √ 3 πœƒ = 0 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

  • A { 1 5 0 , 3 3 0 } ∘ ∘
  • B { 3 0 , 3 3 0 } ∘ ∘
  • C { 1 5 0 , 2 1 0 } ∘ ∘
  • D { 3 0 , 2 1 0 } ∘ ∘

Q11:

Find the set of values satisfying √ 3 πœƒ βˆ’ √ 3 πœƒ + πœƒ = 0 s e c t a n t a n 2 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

  • A { 1 2 0 , 2 4 0 } ∘ ∘
  • B { 6 0 , 2 4 0 } ∘ ∘
  • C { 6 0 , 3 0 0 } ∘ ∘
  • D { 1 2 0 , 3 0 0 } ∘ ∘

Q12:

Simplify 1 + πœƒ c o t 2 .

  • A c o s 2 πœƒ
  • B βˆ’ πœƒ c s c 2
  • C βˆ’ πœƒ c o s 2
  • D c s c 2 πœƒ

Q13:

Simplify s e c t a n 2 2 πœƒ βˆ’ πœƒ .

  • A c o s 2 πœƒ
  • B βˆ’ 1
  • C βˆ’ πœƒ c o s 2
  • D1

Q14:

Simplify t a n 2 πœƒ + 1 .

  • A s i n 2 πœƒ
  • B βˆ’ πœƒ s e c 2
  • C βˆ’ πœƒ s i n 2
  • D s e c 2 πœƒ

Q15:

Simplify c o t s i n c o s 2 2 2 πœƒ + πœƒ + πœƒ .

  • A s i n 2 πœƒ
  • B βˆ’ πœƒ c s c 2
  • C βˆ’ πœƒ s i n 2
  • D c s c 2 πœƒ

Q16:

Simplify s e c s i n c o s 2 2 2 πœƒ βˆ’ πœƒ + πœƒ .

  • A s i n 2 πœƒ
  • B βˆ’ πœƒ t a n 2
  • C βˆ’ πœƒ s i n 2
  • D t a n 2 πœƒ

Q17:

Simplify 1 + πœƒ 1 + πœƒ t a n c o t 2 2 .

  • A c o t 2 πœƒ
  • B1
  • C βˆ’ 1
  • D t a n 2 πœƒ

Q18:

Simplify 1 + ( 9 0 βˆ’ πœƒ ) c o t 2 ∘ .

  • A t a n 2 πœƒ
  • B c s c 2 πœƒ
  • C c o t 2 πœƒ
  • D s e c 2 πœƒ

Q19:

Suppose that 1 7 𝛼 βˆ’ 8 = 0 s i n with 0 < 𝛼 < 9 0 ∘ ∘ , and that 𝛽 is the largest angle between 0 ∘ and 3 6 0 ∘ for which 3 𝛽 + 4 = 0 t a n . Find the exact value of c s c c o t s e c t a n ( 1 8 0 + 𝛼 ) ( 9 0 βˆ’ 𝛽 ) βˆ’ ( 3 6 0 + 𝛼 ) ( 3 6 0 βˆ’ 𝛽 ) ∘ ∘ ∘ ∘ .

  • A βˆ’ 3 9 1 9 0
  • B βˆ’ 1 1 9 9 0
  • C 3 9 1 9 0
  • D 1 1 9 9 0

Q20:

Find the value of t a n c o t 3 3 πœƒ + πœƒ given t a n c o t πœƒ + πœƒ = 1 6 .

Q21:

𝐴 𝐡 𝐢 is a triangle where t a n 𝐴 = 1 4 and 𝐡 = 2 𝐴 . Find s i n 𝐢 without using a calculator.

  • A √ 1 7 1 7
  • B 5 2 √ 1 7 2 8 9
  • C 8 1 7
  • D 4 7 √ 1 7 2 8 9

Q22:

Simplify ( 1 + πœƒ ) βˆ’ 2 πœƒ c o t c o t 2 .

  • A c o t 2 πœƒ
  • B s e c 2 πœƒ
  • C t a n 2 πœƒ
  • D c s c 2 πœƒ

Q23:

Simplify 1 + ο€» βˆ’ πœƒ  1 + ο€» βˆ’ πœƒ  c o t t a n 2 3 πœ‹ 2 2 πœ‹ 2 .

  • A 1
  • B c o t 2 πœƒ
  • C βˆ’ 1
  • D t a n 2 πœƒ