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If sin⁴x​ + cos⁴x =1/5 2 3 Then what is the value of tan²x=? And sin^8x + cos^8 = ? 8. 27

If  sin⁴x​   +   cos⁴x   =1/5
        2                3
Then what is the value of  tan²x=?
And  sin^8x  +  cos^8  =  ?
            8.                 27

Grade:11

1 Answers

Aarushi Ahlawat
41 Points
3 years ago
Convert the given equation into a whole square in terms of sin^2(x) and constant. Now computation of desired functions is straight forward.
\frac{sin^{4}(x)}{2}+\frac{cos^{4}(x)}{3}=\frac{1}{5}
3sin^{4}(x)+2cos^{4}(x)=\frac{6}{5}
3sin^{4}(x)+2(1-sin^{2}(x))^{2}=\frac{6}{5}
25sin^{4}(x)-20sin^{2}(x)+4=0
(5sin^{2}(x)-2)^{2}=0
sin^{2}(x)=\frac{2}{5}
Hence
 
cos^{2}(x)=\frac{3}{5}
So 
 
tan^{2}(x)=\frac{2}{3}
and
 
\frac{sin^{8}(x)}{8}+\frac{cos^{8}(x)}{27}=(\frac{1}{5})^{3}+(\frac{1}{5})^{3}=\frac{2}{125} 

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