Trigonometric Identities

[greatwhiteshark]

1) Prove the following:

sin(x + y)sin(x - y) = sin"x - sin"z

By using this result or otherwise, prove the following:

sin(x + y + z)sin(y + z - x)sin(z + x - y)sin(x + y - z)

= (a + b + c)(b + c - a)(c + a - b)(a + b - c) - 4a"b"c"

...where a = sin(x), b = sin(y), c = sin(z), and a" = a^2 = "a squared".

2) If cos(x) + cos(3x) = kcos(y) and sin(x) + sin(3x) = ksin(x), show that
cos(x) = ± k/2, and find the values of tan(y) and cos(2y) in terms of k.

 

[arthur ohlsten]

sin[x+y]sin [x-y]=sin^2x - sin^2y

but sin [a+b]=sina cosb + cosa sinb
sin[-a]=-sina
cos[-a] = cos a
cos^2a = 1-sin^2a

[sin x cosy + cosx sin y ][sinxcosy-cosxsiny]=? sin^2x-sin^2y
[sin^2xcos^2y -sinxcosycosxsiny+cosxsinysinxcosy-cos^2xsin^2y]=?
[sin^2xcos^2y - cos^2xsin^2y]=?
[sin^2 x [1-sin^2y] - [1-sin^2x]sin^2y]=?
sin^2x-sin^2x sin^2y- sin^2y+sin^2xsin^2y]=?
sin^2x -sin^2y =? qed

arthur

 

[greatwhiteshark]

okay

Nice steps and easy to follow.