Verifying a Trigonometric Identity...4

[nycmathguy]

Section 5.5

Can you please get me started with 69?

The plus/minus on the numerator throws me into a loop.

 

[MathLover1]

69.

(sin(x)+-sin(y})/(cos(x)+cos(y))=tan((x+-y)/2)

separate in two:

(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)
and
(sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)

prove one:

(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)

manipulate left side

(sin(x)+sin(y))/(cos(x)+cos(y)).....rewrite using trigonometric identities

sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)

cos(x)+cos(y)=2cos((x+y)/2)cos((x−y)/2)

then you have

(sin(x)+sin(y))/(cos(x)+cos(y))

=(2sin((x+y)/2)cos((x-y)/2))/2cos((x+y)/2)cos((x−y)/2)........simplify

=sin((x+y)/2))/cos((x+y)/2)...........use identity tan(x)=sin(x)/cos(x)

=tan((x+y)/2))

same is for (sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)

 

[nycmathguy]

69.

(sin(x)+-sin(y})/(cos(x)+cos(y))=tan((x+-y)/2)

separate in two:

(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)
and
(sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)

prove one:

(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)

manipulate left side

(sin(x)+sin(y))/(cos(x)+cos(y)).....rewrite using trigonometric identities

sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)

cos(x)+cos(y)=2cos((x+y)/2)cos((x−y)/2)

then you have

(sin(x)+sin(y))/(cos(x)+cos(y))

=(2sin((x+y)/2)cos((x-y)/2))/2cos((x+y)/2)cos((x−y)/2)........simplify

=sin((x+y)/2))/cos((x+y)/2)...........use identity tan(x)=sin(x)/cos(x)

=tan((x+y)/2))

same is for (sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)

I was going to work it out but I thank you.
Go check out what Country Boy been saying about your replies to my threads in the Algebra forum. In correcting your replies, he has confused me even more.

One more for today:

Trigonometric Equations