cos⁶θ + sin⁶θ

= (cos²θ)³ + (sin²θ)³

= (cos²θ + sin²θ)( (cos²θ)² - cos²θsin²θ + (sin²θ)² )


(Recall the property a³ + b³ = (a + b)(a² - ab + b²) )

= (1)( (cos²θ)² + 2cos²θsin²θ + (sin²θ)² - 3cos²θsin²θ)


(Completing the square, + recall cos²A + sin²A = 1)

= (cos²θ + sin²θ)² - 3(sinθcosθ)²

= 1² - 3(½sin2θ)²

(Recall sin2A = 2sinAcosA, so sinAcosA = ½sin2A)

= 1 - ¾sin²2θ

= (8 - 6 sin²2θ) / 8

= (5 + 3(1 - 2sin²2θ)) / 8

= (5 + 3cos4θ)/8 (proved)

(Recall cos 2A = 1 - 2sin²A = cos²A - sin²A = 2cos²A - 1. Here your A = 2θ)

You actually succeeded in finding the symbol for power 6.

I only know power 2 and 3, but I have no idea how to input higher powers.

Google keyboard for Android has ¹²³⁴ only.

Other superscripts and subscripts can be easily generated online (there are plenty of sites that do this)

You can even copy and paste from the search results.

Eg. Google 'superscript 6'


So it's a matter of effort. Using ^ to represent superscript just makes things less clear and confusing.

Thank you so much, you are really helpful! This helps a lot!