Worksheet: Product-to-Sum Identities

In this worksheet, we will practice using the product-to-sum identity to simplify and evaluating trigonometric expression.

Q1:

Simplify sincoscossin(𝑥+𝑦)𝑦(𝑥+𝑦)𝑦.

  • A2𝑥𝑦sincos
  • Bcos𝑥
  • C0
  • Dsin𝑥

Q2:

Simplify coscos(𝐴𝐵)(𝐴+𝐵).

  • A0
  • B2𝐴𝐵coscos
  • C2𝐴𝐵sincos
  • D2𝐴𝐵sinsin

Q3:

Simplify sinsin(𝐴+𝐵)(𝐴𝐵).

  • A0
  • B2𝐴𝐵sincos
  • C2𝐴𝐵sinsin
  • D2𝐴𝐵cossin

Q4:

Write the expression 4(70)(33)coscos as a sum or difference of trigonometric expressions.

  • A2(37)+2(103)coscos
  • B2(103)2(37)coscos
  • C2103237sinsin
  • Dcoscos(37)+(103)
  • E237+2103sinsin

Q5:

Find the exact value of 2(195)(45)cossin.

  • A1
  • B312
  • C13
  • D13
  • E132

Q6:

Write the expression sinsin(80)+(20) as a product of trigonometric expressions.

  • Asinsin(50)(30)
  • B2(50)(30)sincos
  • Csincos(50)(30)
  • D2(50)(30)coscos
  • E2(50)(30)sinsin

Q7:

Find the value of coscossinsin5𝑋𝑋+5𝑋𝑋 given sin2𝑋=38.

  • A2332
  • B4132
  • C7364
  • D5564

Q8:

Find cos(2𝑎+𝑏) given 4𝑎=3tan and 3𝑏=4tan where 𝑎 and 𝑏 are acute angles.

  • A0
  • B35
  • C117125
  • D310

Q9:

The intensity of an electric current is given by 𝐶=52(75𝑡)sin where 𝑡 is the time in seconds. Find the intensity after one second without using a calculator using sum and product formulae.

  • A562
  • B5628
  • C56+2
  • D56+28

Q10:

Find the value of sinsinsinsinsinsin𝜋12+3𝜋12+5𝜋12+7𝜋12+9𝜋12+11𝜋12.

Q11:

Find the value of sinsinsinsinsinsinsinsin𝜋16+3𝜋16+5𝜋16+7𝜋16+9𝜋16+11𝜋16+13𝜋16+15𝜋16.

Q12:

Expand sin𝜃+𝜋6 using sum and difference formulas.

  • A2𝜃+3𝜃sincos
  • B12𝜃+3𝜃sincos
  • C123𝜃+𝜃sincos
  • D12(𝜃+𝜃)sincos

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