Anyone know a lot about trig identities?

Question

Anyone Know how to do these using identities?
tan255degrees
sin2(22.5)
Verify this Identity
tan^2xsin^2x=tan^2x-sin^2x

Answer 1

1) Tan( 255 degrees )

Let's convert 255 degrees to radians. To do that we multiply be pi/180.
255 * pi/180 = 255pi/180 = 51pi/36 = 17pi/12

We don't know the tan of (17pi/12), so we have to use the appropriate sum formula for tangent to solve this.

Remember that
tan(a + b) = [tan(a) + tan(b)]/[1 - tan(a)tan(b)]

We have to decompose 17pi/12 into two known unit circle values. 17pi/12 = 9pi/12 + 8pi/12, or 3pi/4 + 2pi/3

tan(3pi/4 + 2pi/3) = [tan(3pi/4) + tan(2pi/3)]/[1 - tan(3pi/4)tan(2pi/3)]

= [1 + sqrt(3)]/[1 - (1)sqrt(3)] = [1 + sqrt(3)]/[1 - sqrt(3)]

We can reduce this by multiplying numerator and denominator by the conjugate, [1 + sqrt(3)], giving us

[1 + sqrt(3)][1 + sqrt(3)] / [1 - sqrt(3)][1 + sqrt(3)]

or

[1 + 2sqrt(3) + 3] / [1 - 3]
= 4 + 2sqrt(3) / (-2)
= -2 - sqrt(3)

The rest is homework.

Answer 2

in degrees
tan255 would be the same as sin255/cos255 = 3.73
The second one relates to the double angle identity for sin
sin2(22.5) = 2sin(22.5) cos(22.5)

tan^2 x * sin^2 x = tan^2 x - sin^2 x
You can use identities to solve that, but I don't believe that in and of itself is an identity.

Answer 3

tan255degrees - TOO HOT
sin2(22.5) - GOT ME
Verify this Identity
tan^2xsin^2x=tan^2x-sin^2x - VERIFIED