Penjelasan dengan langkah-langkah:
cos 165°(cos 135° + tan 15°) = cos (180°-15°) (cos (180°-45°) + tan 15°)
= -cos 15° (-cos 45° + tan 15°)
= cos 15° cos 45° - cos 15° tan 15°
= cos (45°-30°) cos 45° - sin 15°
= (cos 45° cos 30° + sin 45° sin 30°) cos 45° - sin (45°-30°)
= ((1/2)√2 · (1/2)√3 + (1/2)√2 · (1/2)) · (1/2)√(2) - (sin 45° cos 30° - cos 45° sin 30°)
= ((1/2)·(1/2)·√(2) ·√(3) + (1/2)·(1/2)·√(2)) ·(1/2)√(2) - ((1/2)√(2) · (1/2)√(3) - (1/2)√(2) · (1/2))
= ((1/4)√(2·3) + (1/4)√(2)) ·(1/2)√(2) - ((1/2)·(1/2)·√(2) ·√(3) - (1/2)·(1/2)·√(2))
= ((1/4)√(6) + (1/4)√(2)) ·(1/2)√(2) - ((1/4)√(2·3) - (1/4)·√(2))
= ((1/4)√(6) + (1/4)√(2)) ·(1/2)√(2) - ((1/4)√(6) - (1/4)·√(2))
= (1/4)√(6)·(1/2)√(2) + (1/4√(2) ·(1/2)√(2) - (1/4)√(6) + (1/4)√(2)
= (1/4)·(1/2)·√(6) ·√(2) + (1/4)·(1/2)·√(2)·√(2) - (1/4)√(6) + (1/4)√(2)
= (1/8)√(6·2) + (1/8)√(2·2) - (1/4)√(6) + (1/4)√(2)
= (1/8)√(12) + (1/8)√(4) - (1/4)√(6) + (1/4)√(2)
= (1/8)√(4·3) + (1/8)·2 - (1/4)√(6) + (1/4)√(2)
= (1/8)·√(4) ·√(3) + 2/8 - (1/4)√(6) + (1/4)√(2)
= (1/8)·2·√(3) + (1/4) - (1/4)√(6) + (1/4)√(2)
= (2/8)√(3) + (1/4) - (1/4)√(6) + (1/4)√(2)
= (1/4)√(3) + (1/4) - (1/4)√(6) + (1/4)√(2)
Kesimpulan:
Jadi nilai dari cos 165°(cos 135° + tan 15°) = (1/4)√(3) + (1/4) - (1/4)√(6) + (1/4)√(2)
