Use complementary angles (cos(90∘−θ)=sin‌θ) to convert some cosines to sines, then use product-to-sum formulas to simplify the product. Finally, express everything in terms of cos‌10∘ and compare with the given form α+‌
√3
16
‌cos‌10∘ to find α.
Start with E=cos2(10∘)‌cos(20∘)‌cos(40∘)‌cos(50∘)‌cos(70∘). Use cos‌50∘=sin‌40∘,cos‌70∘=sin‌20∘ : E=cos210∘⋅cos‌20∘‌cos‌40∘‌s‌i‌n‌40∘sin‌20∘. Group: sin‌20∘‌cos‌20∘=‌
1
2
sin‌40∘,‌‌sin‌40∘‌cos‌40∘=‌
1
2
sin‌80∘. So E=cos210∘⋅‌
1
2
sin‌40∘⋅‌
1
2
sin‌80∘=‌
1
4
cos210∘sin‌40∘sin‌80∘. Use 2sin‌Asin‌B=cos(A−B)−cos(A+B) : sin‌40∘sin‌80∘=‌
1
2
(cos‌40∘−cos‌120∘)=‌
1
2
(cos‌40∘+‌
1
2
). So E=‌
1
4
cos210∘⋅‌
1
2
(cos‌40∘+‌
1
2
)=‌
1
8
cos210∘(cos‌40∘+‌
1
2
). Now, using multiple-angle identities (expressing cos‌40∘ in terms of cos‌10∘ ) and the fact that cos‌30∘=cos(3⋅10∘)=‌
√3
2
, this expression simplifies to the required form: E=α+‌