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Given (sinθ + cosθ) / sinθcosθ = 1/3. What is the value of sin⁴θ – cos⁴θ?

Given
(sinθ + cosθ) / sinθcosθ = 1/3.
What is the value of sin⁴θ – cos⁴θ? 

Grade:10

1 Answers

Aditya Gupta
2081 Points
4 years ago
actually sayan i m pretty sure u have typed the ques incorrectly. but i ll still tell u the method to even solve this.
let sin theta= s and cos theta= c
so sin⁴θ – cos⁴θ= (s^2 – c^2)(s^2 + c^2)= (s^2 – c^2)(1)= s^2 – c^2= (s – c)(s+c)......(*)
now, let s+c= x
sc= y
and s – c= z
now, we see that x^2= s^2 +  c^2 + 2sc= 1+2y......(1)
also, given that x/y= 1/3
or y= 3x.....(2)
from (1) and (2) eliminating y, we have
x^2= 1+6x
or x^2 – 6x – 1= 0
x= 3 ± sqrt 10, but we know that the range of s+c is from – sqrt 2 to sqrt 2.
hence, x= 3 – sqrt 10
now, x^2 + z^2= 2
or z= ± sqrt (2 – x^2)
from (*), we need to find the value of xz. since both x and z are known, we can obtain the final answer.
kindly approve :)

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