Question

Find the value of:
(cos^4(75°) + sin^4(75°) + 3sin^2(75º)cos^2(75°)) / (cos^6(75º) + sin^6(75°) + 4sin^2(75°)cos^2 (75º))

Answer 1

uhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh what? i cant understand dis

Answer 2

I first found the values for sin75 and cos75. (skipping the degree symbol for easier typing)
sin75 = sin(30+45) = sin30cos45 + cos30sin45 = (1/2)(√2/2) + (√3/2)(√2/2) = ( √2 + √6 /4)
sin^2 75 = (√2+√6)^2/16 = (4 + 2√12 + 6)/16 = (8 + 4√3)/16 = (2 + √3)/4
sin^4 75 = [ (2+√3)/4 ]^2 = .... = (7 + 4√3)/16
sin^6 75 = (sin^2 75)(sin^4 75) = (2 + √3)/4 * (7 + 4√3)/16 = (26 + 15√3)/64

I then did the same for cos75 = cos30cos45 - sin30sin45 = (√3/2)(√2/2) - (1/2)(√2/2) = (√6 - √2)/4
cos^2 75 = [ (√6-√2)/4 ]^2 = (6 - 2√12 + 2)/16 = (8 - 4√3)/16 = (2 - √3)/4
cos^4 75 = [ (2 - √3)/4 )^2 = ....... = (7 - 4√3)/16
cos^6 75 = (cos^2 75)(cos^4 75) = [(2 + √3)/4][(2 - √3)/4] = (26-15√3)/64

and finally (sin^2 75)(cos^2 75) = [(2 + √3)/4][(2 - √3)/4) = 1/16

(notice the nice symmetry between the corresponding answers, that is useful,

almost done .....

(cos^4(75°) + sin^4(75°) + 3sin^2(75º)cos^2(75°)) / (cos^6(75º) + sin^6(75°) + 4sin^2(75°)cos^2 (75º))
= [ (7 - 4√3)/16 + (7 + 4√3)/16 + 3(1/16) ] / [(26-15√3)/64 + (26 + 15√3)/64 + 4(1/16 ]
= [ 17/16] / [68/64]
= (17/16)(64/68)
= 1

yeaahhhhh!!!! That was fun

I skipped several of the simplifications as you can see, however, they are all correct
I will leave it up to you to verify the missing ones.

Answer 3

(cos^4x + sin^4x + 3 sin^2x cos^2x)
--------------------------------------------------
(cos^6x + sin^6x + 4 sin^2x cos^2x)

= (-cos^4x + cos^2x + 1) / (-cos^4x + cos^2x + 1)
= 1