Worksheet: Trigonometric Identities

Practice

In this worksheet, we will practice simplifying trigonometric expressions by applying trigonometric identities.

Q1:

Simplify s i n c o s c s c c o t 2 2 2 2 𝜃 + 𝜃 𝜃 − 𝜃 .

  • A c o s 2 𝜃
  • B − 1
  • C − 𝜃 c o s 2
  • D1

Q2:

Simplify ( 1 − 𝜃 ) + ( 1 + 𝜃 ) t a n t a n 2 2 .

  • A s e c 2 𝜃
  • B 2 𝜃 c s c 2
  • C c s c 2 𝜃
  • D 2 𝜃 s e c 2

Q3:

Knowing that 5 + 4 𝑥 = − 1 2 𝑥 c o s t a n 2 , find t a n 𝑥 .

  • A t a n 𝑥 = − 1 2
  • B t a n 𝑥 = 3 2
  • C t a n 𝑥 = − 5 2
  • D t a n 𝑥 = − 3 2
  • E t a n 𝑥 = 5 2

Q4:

Simplify 1 − 2 𝑥 1 + 2 𝑥 c o s c o s .

  • A t a n 2 𝑥
  • B c o t 2 𝑥
  • C c o t 𝑥
  • D t a n 𝑥

Q5:

Simplify 1 + ( 9 0 − 𝜃 ) c o t 2 ∘ .

  • A t a n 2 𝜃
  • B c s c 2 𝜃
  • C c o t 2 𝜃
  • D s e c 2 𝜃

Q6:

Simplify ( 1 + 𝜃 ) − 2 𝜃 c o t c o t 2 .

  • A c o t 2 𝜃
  • B s e c 2 𝜃
  • C t a n 2 𝜃
  • D c s c 2 𝜃

Q7:

Simplify 1 + 𝜃 1 + 𝜃 t a n c o t 2 2 .

  • A c o t 2 𝜃
  • B1
  • C − 1
  • D t a n 2 𝜃

Q8:

Simplify 1 + − 𝜃 1 + − 𝜃 c o t t a n 2 3 𝜋 2 2 𝜋 2 .

  • A 1
  • B c o t 2 𝜃
  • C − 1
  • D t a n 2 𝜃

Q9:

Find the set of values satisfying s e c t a n t a n 2 2 𝜃 − 𝜃 + √ 3 𝜃 = 0 where 0 ≤ 𝜃 < 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 } ∘ ∘

Q10:

Simplify 1 + 𝜃 c o t 2 .

  • A c o s 2 𝜃
  • B − 𝜃 c s c 2
  • C − 𝜃 c o s 2
  • D c s c 2 𝜃

Q11:

Find the value of t a n c o t ( 𝜋 + 𝐴 ) − 𝐴 − 𝜋 2 given 2 1 𝐴 = − 2 9 c s c where 3 𝜋 2 < 𝐴 < 2 𝜋 .

  • A 4 1 4 2 0
  • B 2 1 1 0
  • C − 4 1 4 2 0
  • D − 2 1 1 0

Q12:

Find the possible values of t a n c o t 2 2 𝜃 − 𝜃 given that t a n c o t 𝜃 + 𝜃 = 2 4 .

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

Q13:

Find the value of t a n c o t 𝜃 + 𝜃 given t a n c o t 𝜃 + 𝜃 = 1 6 .

Q14:

Suppose that 1 7 𝛼 − 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 − 𝛽 ) − ( 3 6 0 + 𝛼 ) ( 3 6 0 − 𝛽 ) ∘ ∘ ∘ ∘ .

  • A − 3 9 1 9 0
  • B − 1 1 9 9 0
  • C 3 9 1 9 0
  • D 1 1 9 9 0

Q15:

𝐴 𝐵 𝐶 is a triangle where t a n 𝐴 = 1 4 and 𝐵 = 2 𝐴 . Find s i n 𝐶 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

Q16:

Find s i n ( 2 𝑎 + 𝑏 ) given t a n 𝑎 = 3 and t a n 𝑏 = 2 where 𝑎 and 𝑏 are acute angles.

  • A − 1 7 √ 5 5 0
  • B − 7 √ 5 2 5
  • C − 1 3 √ 5 5 0
  • D − √ 5 5

Q17:

Find all the possible values of s i n c o s c o s s i n 3 𝑋 𝑋 + 3 𝑋 𝑋 given that s i n 2 𝑋 = 4 7 .

  • A √ 3 3 7 , − √ 3 3 7
  • B 1 7 4 9
  • C 4 7
  • D 8 √ 3 3 4 9 , − 8 √ 3 3 4 9

Q18:

Simplify 5 𝑎 𝑎 s i n c o t 2 .

  • A 5 2 2 𝑎 c o s
  • B s i n 2 𝑎
  • C c o s 2 𝑎
  • D 5 2 2 𝑎 s i n

Q19:

Simplify √ 𝑎 𝑎 c o s s i n 2 2 .

  • A 1 2 2 𝑎 c o s
  • B s i n 2 𝑎
  • C c o s 2 𝑎
  • D 1 2 2 𝑎 s i n

Q20:

Simplify 1 − 𝜃 𝜃 − 𝜃 s i n c s c c o t 2 2 2 .

  • A s i n 2 𝜃
  • B − 𝜃 c o s 2
  • C − 𝜃 s i n 2
  • D c o s 2 𝜃

Q21:

Simplify s e c s e c t a n 2 2 2 𝜃 − 1 𝜃 − 𝜃 .

  • A s i n 2 𝜃
  • B − 𝜃 t a n 2
  • C − 𝜃 s i n 2
  • D t a n 2 𝜃

Q22:

Find the set of values satisfying s e c t a n t a n 2 2 𝜃 − 𝜃 − 𝜃 = 0 where 0 ≤ 𝜃 < 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 } ∘ ∘

Q23:

Find the set of values satisfying s e c t a n t a n 2 2 𝜃 − 𝜃 − √ 3 𝜃 = 0 where 0 ≤ 𝜃 < 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 } ∘ ∘

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