Question

Given: Cosine (x minus y)

Equals sine left-brace StartFraction pi Over 2 EndFraction minus (x minus y) right-brace
Equals sine left-brace StartFraction pi Over 2 EndFraction minus x + y right-brace
Equals sine left-brace (StartFraction pi Over 2 EndFraction minus x) + y right-brace
Equals sine left-brace (StartFraction pi Over 2 EndFraction minus x) minus (negative y) right-brace
Equals sine (StartFraction pi Over x EndFraction minus x) cosine (negative y) minus cosine (StartFraction pi Over 2 EndFraction minus x) sine (negative y)
Equals cosine (x) cosine (negative y) minus sine (x) sine (negative y)
Equals cosine (X) cosine (y) + sine (x) sine (y)
Choose a justification for each step in the derivation of the sine difference identity.

Step 1:

Step 2: Distributive property

Step 3: Associative property

Step 4: Factoring out –1

Step 5:

Step 6:

Step 7:

Answer 1

The required steps that complete the proof are:

Step 1: Cofunction Identity

Step 5: Sine Difference Identity

Step 6: Cofunction Identity

Step 7: Cosine Function Is Even, Sine Function Is Odd

Trigonometry identity

Given the cosine function expressed as:

• cos(x - y)

The first step is to apply the cofunction identity as shown:

• Step 1: Cofunction Identity

cos(90 - x) = sin x

cos(x - y) = sin (90-(x-y))

• Step 5: Sine Difference Identity

For the fifth step, you will apply the sine difference identity.

• Step 6: Cofunction Identity

To get the sixth step, you will apply the Cofunction Identity on the result in the fifth step.

The final step 7 will be

• Step 7: Cosine Function Is Even, Sine Function Is Odd

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