Question

सिद्ध कीजिए। 

sin⁴θ - cos⁴θ = 1 - 2cos²θ 

Answer 1

बायाँ पक्ष = sin⁴θ - cos⁴θ

= `(sin^2theta)^2 - (cos^2theta)^2`

= `(sin^2theta + cos^2theta)(sin^2theta - cos^2theta)` ..........`[a^2 - b^2 = (a + b)(a - b)]`

= `1(sin^2theta - cos^2theta)` ................`(sin^2theta + cos^2theta = 1)`

= `sin^2theta - cos^2theta`

= `1 - cos^2theta - cos^2theta` ........`[(∵ sin^2theta + cos^2theta = 1),(∴ sin^2theta = 1 - cos^2theta)]`

= `1 - 2cos^2theta`

= दायाँ पक्ष

∴ बायाँ पक्ष = दायाँ पक्ष

∴ sin⁴θ - cos⁴θ = 1 - 2cos²θ.