Final answer:
To compute sin⁴x+cos⁴ x, we can use the identity (sinx+cosx)² = sin²x + 2sinxcosx + cos²x. Simplifying further, we have sin⁴x + cos⁴x = (4/25)² - 4/25(sinxcosx).
Step-by-step explanation:
To compute sin⁴x+cos⁴ x, we can use the identity (sinx+cosx)² = sin²x + 2sinxcosx + cos²x. Rearranging this equation gives us sin²x + cos²x = (sinx+cosx)² - 2sinxcosx. We are given that sinx+cosx = 2/5. Substituting this into the equation, we get sin²x + cos²x = (2/5)² - 2(sinxcosx). Simplifying further, we have sin²x + cos²x = 4/25 - 2(sinxcosx).
Squaring this equation again, we get sin⁴x + 2sin²xcos²x + cos⁴x = (4/25)² - 4/25(sinxcosx). Since 2sin²xcos²x = (sin²x)(cos²x), we have sin⁴x + cos⁴x = (4/25)² - 4/25(sinxcosx) as our answer.