The missing reasons are Cofunction identity, Cosine difference identity, Cofunction identity, Cosine function is even and sine function is odd respectively.
Given:
The given expression is [tex]\sin (x-y)[/tex].
To find:
The justification for each step.
Explanation:
Cofunction identities:
[tex]\sin\left(\dfrac{\pi}{2}-x\right)=\cos x[/tex]
[tex]\cos\left(\dfrac{\pi}{2}-x\right)=\sin x[/tex]
Cosine difference identity:
[tex]\cos(x-y)=\cos x\cos y+\sin x\sin y[/tex]
Cosine function is even, so [tex]\cos(-x)=\cos x[/tex].
Sine function is odd, so [tex]\sin(-x)=-\sin x[/tex].
We have,
[tex]\sin (x-y)[/tex]
[tex]=\cos\left|\dfrac{\pi}{2}-(x-y)\right|[/tex] Step 1: Cofunction identity
[tex]=\cos\left|\dfrac{\pi}{2}-x+y\right|[/tex] Step 2: Distributive property
[tex]=\cos\left|\left(\dfrac{\pi}{2}-x\right)+y\right|[/tex] Step 3: Associative property
[tex]=\cos\left|\left(\dfrac{\pi}{2}-x\right)-(-y)\right|[/tex] Step 4: Factoring out
[tex]=\cos\left(\dfrac{\pi}{2}-x\right)\cos(-y)+\sin\left(\dfrac{\pi}{2}-x\right)\sin(-y)[/tex] Step 5: Cosine difference identity
[tex]=\sin(x)\cos(-y)+\cos(x)\sin(-y)[/tex] Step 6: Cofunction identity
[tex]=\sin(x)\cos(y)-\cos(x)\sin(y)[/tex] Step 7: Cosine function is even, sine function is odd
Therefore, the missing reasons are Cofunction identity, Cosine difference identity, Cofunction identity, Cosine function is even, sine function is odd respectively.
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