Question

Review the incomplete derivation of the cosine sum identity.

A 2-column table with 5 rows. Column 1 has entries step 1, step 2, step 3, step 4, step 5. Column 2 has entries cosine (x + y), sine (StartFraction pi Over 2 EndFraction minus (x + y) ), blank, sine (StartFraction pi Over 2 EndFraction minus x) cosine (negative y) + cosine (StartFraction pi Over 2 EndFraction minus x) sine (negative y), blank.

Which expressions for Step 3 and Step 5 complete the derivation?

Step 3: Sine ( (StartFraction pi over 2 EndFraction minus x) + y )
Step 5: cos(x)cos(y) + sin(x)sin(y)
Step 3: Sine ( (StartFraction pi over 2 EndFraction minus x) + y )
Step 5: cos(x)cos(y) – sin(x)sin(y)
Step 3: Sine ( (StartFraction pi over 2 EndFraction minus x) minus y )
Step 5: cos(x)cos(y) + sin(x)sin(y)
Step 3: Sine ( (StartFraction pi over 2 EndFraction minus x) minus y )
Step 5: cos(x)cos(y) – sin(x)sin(y)

Answer 1

D

Explanation:

Top Answer was right, don't know why it was rated poorly

Answer 2

Option (4)

Explanation:

STEP - 1

cos(x + y)

STEP - 2

`\text{sin}[(\pi)/(2)-(x+y)]`

STEP - 3

`\text{sin}[((\pi)/(2)-x)-y]`

STEP - 4

`\text{sin}((\pi)/(2)-x)\text{cos}(-y)+\text{cos}((\pi)/(2)-x)\text{sin}(-y)`

STEP - 5

cos(x)cos(y) - sin(x)sin(y)

[Since,
`\text{sin}((\pi)/(2)-x)=cos(x)` and
`\text{cos}((\pi)/(2)-x)=\text{sin}(x)`]

[Since, cos(-x) = cos(x) and sin(-x) = -sin(x)]

Therefore, Option (4) will be the correct option.