(i) sin (27θ° + θ) → IV quardant, -ive = -cosθ … (i)
sin (27θ° – θ) → III quardant, -ive = – cos θ …. (ii)
cos (18θ° + θ) → III quardrant, -ive = – cos θ … (iii)
L.H.S. = cos θ + sin (27θ° + θ) – sin (27θ° – θ) + cos (18θ° + 9)
= cos θ – cos θ + cos θ – cos θ [from (i), (ii) and (iii)]
= θ
= R.H.S.
Hence Proved.
(ii) LHS = sec(3π/2 - θ)sec(θ - 5π/2) + tan(5π/2 + θ)tan(θ - 3π/2)
= sec (27θ° – θ) sec (θ – 45θ°)+ tan (45θ° + θ) tan (θ – 27θ°)
= – cosec θ sec (θ – 45θ°)+ tan (36θ° + 9θ° + θ) tan (θ – 27θ°)
= – cosec θ sec (45θ° – θ) – tan (9θ° + θ) tan (27θ° – θ)
= – cosec θ sec (36θ° + 9θ° – θ) – (- cot θ) cot θ
= – cosec θ sec (9θ° – θ) + cot2 θ
= – cosec θ cosec θ + cot2 θ
= – cosec2 θ + cot2 θ
= – 1 (∵ 1 + cot2 θ = cosec2 θ cot2 θ – cosec2 θ = – 1)
R.H.S.
Hence Proved.