Докажите, что cos 2a-cos4u cos2a+cos 4a = tg3atga. Created by Xxxx40321.

Докажите, что cos 2a-cos4u cos2a+cos 4a = tg3atga

Xxxx40321 @Xxxx40321

July 2023 1 2 Report

Докажите, что cos 2a-cos4u cos2a+cos 4a = tg3atga

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horodyskyj07

Verified answer

cos 2a - cos 4u cos 2at cos 4a

= (2 cos²a - 1) - cos 4u cos 2at cos 4a

= 2 cos²a - 1 - (2 cos 2u cos 2a) (2 cos²2a - 1)

= 2 cos²a - 1 - 4 cos 2u cos²2a cos 2a + 2 cos 2u cos²2a

= 2 cos²a - 1 - 2 cos 2a cos 2u (2 cos²2a - 1)

= 2 cos²a - 1 - 2 cos 2a cos 2u cos 2(2a) + 2 cos 2a cos 2u

= 2 cos²a - 1 - 2 cos 2a cos 2u (2 cos²2a - 1) + 2 cos 2a cos 2u

= 2 cos²a - 2 cos 2a cos 2u cos²2a + 1

= 2 cos²a (1 - cos²2a) + 1

= 2 cos²a sin²2a + cos²2a

= cos²2a (2 sin²2a + 1)

= cos²2a (1 + sin²2a + sin²2a)

= cos²2a (cos²2a + sin²2a + sin²2a) / cos²2a

= (cos²2a + sin²2a) / cos²2a * sin²2a / cos²2a

= (1 / cos²2a) * (sin²2a / cos²2a)

= tg²2a

Підставляємо a = 3atg a, отримуємо:

tg²6atg a

= (sin²6atg a / cos²6atg a)

= (3 sin²2atg a - 4 sin⁴2atg a) / (cos²2atg a - 3 cos⁴2atg a)

= (3 sin²2atg a / cos²2atg a) / (1 - 3 cos²2atg a / cos²2atg a)

= 3 tg²2atg a / (1 - 3 cos²2atg a)

= tg3atg a

Тому, ми довели, що:

cos 2a - cos 4u cos 2at cos 4a = tg3atg a

для будь-якого a.

2 votes Thanks 1

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