Simplify ‘4*(cos 42^(@) cos 102^(@) cos 114^(@) cos 174^(@))^(2)’ into a rational number.​

1 thought on “Simplify ‘4*(cos 42^(@) cos 102^(@) cos 114^(@) cos 174^(@))^(2)’ into a rational number.​”

1. 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) .

   Learn more :-

   It sino + tano = m

   tano – sino an

   Then express the

   values of m²-n² in terms

   of M and N

   https://brainly.in/question/13926306

   tanA/(1-cotA) + cotA/(1-tanA)

   https://brainly.in/question/16775946

   Reply