Simplify '4*(cos 42^(@) cos 102^(@) cos 114^(@) cos 174^(@))^(2)' into a rational number.
Answers
Simplify :- 4(cos 42° * cos 102° * cos 114° * cos 174°)² into a rational number .
Answer :-
→ 4(cos 42° * cos 102° * cos 114° * cos 174°)²
using,
- cos (90° - θ) = sin θ
- cos (90° + θ) = - sin θ
→ 4[sin 48° * (- sin 12°) * (- sin 24°) * ( - sin 84°)]²
→ 4[(-1) * sin 12° * sin 24° * sin 48° * sin 84° ]² --------- Eqn.(1)
solving in parts,
→ sin 12° * sin 48°
multiply and divide by sin 72°
→ (1/sin 72°)[ sin 12° * sin 48° * sin 72°]
using,
- sin A * sin (60 - A) * sin (60 + A) = (1/4) sin 3A,
→ (1/sin 72°)(1/4)sin 36° ------ Eqn.(2)
similarly,
→ sin 24° * sin 84°
multiply and divide by sin 36°
→ (1/sin 36°)[ sin 24° * sin 36° * sin 84°]
→ (1/sin 36°)(1/4)sin 72°------- Eqn.(3)
multiply Eqn.(2) and Eqn.(3) ,
→ {(1/sin 72°)(1/4)sin 36°} * (1/sin 36°)(1/4)sin 72°
→ (1/4) * (1/4) * (sin 36°/sin 36°) * (sin 72°/sin 72°)
→ (1/16)
putting this value in Eqn.(1)
→ 4[(-1) (1/16)]²
→ 4 * (1/256)
→ (1/64) (Ans.)
Hence, our required rational number is (1/64) .
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