a) (sin 25⁰ + cos 25⁰)(sin 25⁰ - cos 25⁰) + 2 sin² 65⁰ = = sin²25⁰ - cos²25⁰ + 2 sin²(90⁰ - 25⁰) = = sin²25⁰ - cos²25⁰ + 2 * cos²25⁰ = sin²25⁰ + cos²25⁰ = 1 b) cos²10⁰ + cos²80⁰ = cos²(90⁰ - 80⁰) + cos²80⁰ = = sin²80⁰ + cos²80⁰ = 1 c) (cos 17⁰ + cos 73⁰)² + ( sin 17⁰ - sin 73⁰) ² = = cos²17⁰ + 2 sin 17⁰ * cos 73⁰ + cos²73⁰ + + sin²17⁰ - 2 sin 17⁰*sin 73⁰ + sin² 73⁰ = = (sin²17⁰ + cos²17⁰) + (sin²73⁰ + cos²73⁰) + 2 sin17⁰cos17⁰ - - 2 sin 17⁰ cos17⁰ = 1 = 1 = 2 bo cos 73⁰ = cos(90⁰ - 17⁰) = sin 17⁰ sin 73⁰ = sin(90⁰ - 17⁰) = cos 17⁰ d) ctg 80⁰ = ctg(90⁰ - 10⁰) = tg 10⁰ ctg 70⁰ = ctg(90⁰- 20⁰) = tg 20⁰ analogicznie ctg 60⁰ = tg 30⁰ ctg 50⁰ = tg 40⁰ zatem ctg 10⁰ *ctg20⁰ *ctg30⁰ *ctg40⁰ * ctg50⁰*ctg60⁰*ctg70⁰*ctg80⁰ = = ctg10⁰ *ctg80⁰*ctg20⁰*ctg70⁰*ctg30⁰*ctg60⁰*ctg40⁰*ctg50⁰ = = ctg10⁰*tg10⁰*ctg20⁰*tg20⁰*ctg30⁰*tg30⁰*ctg40⁰*tg40⁰ = = 1 * 1* 1 * 1 = 1 bo ctg α * tg α = 1 -------------------------------------------------------------------------
