![sin^2 \alpha +sin^2 \beta +cos( \alpha + \beta )cos( \alpha - \beta )= \\ \\ =sin^2 \alpha +sin^2 \beta + \frac{1}{2}(cos( \alpha + \beta + \alpha - \beta )+cos( \alpha + \beta - \alpha + \beta ))= \\ \\ =sin^2 \alpha +sin^2 \beta + \frac{ 1}{2}(cos2 \alpha +cos2 \beta )= \\ \\ =sin^2 \alpha +sin^2 \beta + \frac{1}{2}(cos^2 \alpha -sin^2 \alpha +cos^2 \beta -sin^2 \beta )= \\ \\ ](https://tex.z-dn.net/?f=sin%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2Bcos%28+%5Calpha+%2B+%5Cbeta+%29cos%28+%5Calpha+-+%5Cbeta+%29%3D+%5C%5C++%5C%5C+%0A%3Dsin%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2B+%5Cfrac%7B1%7D%7B2%7D%28cos%28+%5Calpha+%2B+%5Cbeta+%2B+%5Calpha+-+%5Cbeta+%29%2Bcos%28+%5Calpha+%2B+%5Cbeta+-+%5Calpha+%2B+%5Cbeta+%29%29%3D+%5C%5C++%5C%5C+%0A%3Dsin%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2B+%5Cfrac%7B+1%7D%7B2%7D%28cos2+%5Calpha+%2Bcos2+%5Cbeta+%29%3D+%5C%5C++%5C%5C+%0A%3Dsin%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2B+%5Cfrac%7B1%7D%7B2%7D%28cos%5E2+%5Calpha+-sin%5E2+%5Calpha+%2Bcos%5E2+%5Cbeta+-sin%5E2+%5Cbeta+%29%3D+%5C%5C++%5C%5C+%0A)
![=sin^2 \alpha +sin^2 \beta + \frac{1}{2}cos^2 \alpha - \frac{1}{2}sin^2 \alpha + \frac{1}{2} cos^2 \ \beta - \frac{1}{2}sin^2 \beta = \\ \\ = \frac{1}{2}sin^2 \alpha + \frac{1}{2}cos^2 \alpha + \frac{1}{2}sin^2 \beta + \frac{1}{2}cos^2 \beta = \\ \\ = \frac{1}{2}(sin^2 \alpha +cos^2 \alpha +sin^2 \beta +cos^2 \beta )= \\ \\ = \frac{1}{2}(1+1)= \frac{1}{2}*2=1](https://tex.z-dn.net/?f=%3Dsin%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2B+%5Cfrac%7B1%7D%7B2%7Dcos%5E2+%5Calpha+-+%5Cfrac%7B1%7D%7B2%7Dsin%5E2+%5Calpha+%2B+%5Cfrac%7B1%7D%7B2%7D+cos%5E2+%5C+%5Cbeta++-+%5Cfrac%7B1%7D%7B2%7Dsin%5E2+%5Cbeta+%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B1%7D%7B2%7Dsin%5E2+%5Calpha+%2B+%5Cfrac%7B1%7D%7B2%7Dcos%5E2+%5Calpha++%2B+%5Cfrac%7B1%7D%7B2%7Dsin%5E2+%5Cbeta+%2B+%5Cfrac%7B1%7D%7B2%7Dcos%5E2+%5Cbeta+%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B1%7D%7B2%7D%28sin%5E2+%5Calpha+%2Bcos%5E2+%5Calpha+%2Bsin%5E2+%5Cbeta+%2Bcos%5E2+%5Cbeta+%29%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B1%7D%7B2%7D%281%2B1%29%3D+%5Cfrac%7B1%7D%7B2%7D%2A2%3D1++++++++++++)
1=1
Что и требовалось доказать.
P.S.
Используемые формулы:
1) sin²α+cos²α=1
2) cos2α=cos²α - sin²α
3) cosα*cosβ= ¹/₂ (cos(α+β)+cos(α-β))
5,2*100 000=520 000 см = 5200 м = 5,2 км
1см = 100 000 см
1см = 1 000 м
1см = 1 км
Решение смотри в приложении
Ответ 8