cos 75^0 . sec (-15^0) .tan 165/sec 345^0 . sin 195^0 . cot

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Pembahasan

Kita perlu menghitung nilai dari ekspresi: cos 75^0 . sec (-15^0) .tan 165 / (sec 345^0 . sin 195^0 . cot 75)

Mari kita sederhanakan setiap fungsi trigonometri:

*   cos 75° = cos (45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
*   sec (-15°) = sec (15°) = 1 / cos (15°)
    cos 15° = cos (45° - 30°) = cos 45° cos 30° + sin 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
    Jadi, sec (-15°) = 4 / (√6 + √2) = 4(√6 - √2) / (6 - 2) = √6 - √2
*   tan 165° = tan (180° - 15°) = -tan 15°
    tan 15° = sin 15° / cos 15° = [(√6 - √2)/4] / [(√6 + √2)/4] = (√6 - √2) / (√6 + √2) = (√6 - √2)^2 / (6 - 2) = (6 - 2√12 + 2) / 4 = (8 - 4√3) / 4 = 2 - √3
    Jadi, tan 165° = -(2 - √3) = √3 - 2
*   sec 345° = sec (-15°) = sec (15°) = √6 - √2
*   sin 195° = sin (180° + 15°) = -sin 15°
    sin 15° = sin (45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
    Jadi, sin 195° = -(√6 - √2)/4
*   cot 75° = 1 / tan 75°
    tan 75° = tan (45° + 30°) = (tan 45° + tan 30°) / (1 - tan 45° tan 30°) = (1 + 1/√3) / (1 - 1/√3) = (√3 + 1) / (√3 - 1) = (√3 + 1)^2 / (3 - 1) = (3 + 2√3 + 1) / 2 = (4 + 2√3) / 2 = 2 + √3
    Jadi, cot 75° = 1 / (2 + √3) = (2 - √3) / (4 - 3) = 2 - √3

Sekarang substitusikan kembali nilai-nilai ini ke dalam ekspresi:
[cos 75° . sec (-15°) . tan 165°] / [sec 345° . sin 195° . cot 75°]
= [((√6 - √2)/4) * (√6 - √2) * (√3 - 2)] / [(√6 - √2) * (-(√6 - √2)/4) * (2 - √3)]

Perhatikan bahwa (√6 - √2) di pembilang dan penyebut saling menghilangkan.
Juga, (√3 - 2) = -(2 - √3).

= [((√6 - √2)/4) * (√3 - 2)] / [(-(√6 - √2)/4) * (2 - √3)]
= [((√6 - √2)/4) * -(2 - √3)] / [(-(√6 - √2)/4) * (2 - √3)]

Semua suku saling menghilangkan, menyisakan:
= -1 / -1 = 1