If tan (3x) = 5 tan (x), what is the value of sin (x)?

ans. sin (x)=√2/4
solns:
tan(3x)=5tan(x) -----> eq. 1
tan(3x)=tan (2x + x)= (tan (2x)+tan(x))/(1-tan(2x)tan(x))-----> eq. 2
subs. 2 in 1
(tan (2x)+tan(x))/(1-tan(2x)tan(x))=5tan(x)
tan(2x) + tan (x)=5tan(x)-5tan(2x)tan²(x)
tan(2x)=4tan(x)-5tan(2x)tan²(x)
tan(2x)+5tan(2x)tan²(x)=4tan(x)
tan(2x)(1 + 5tan²(x)=4tan(x)----- eq. 3

tan(2x)=2tan(x)/(1-tan²(x)) -----eq.4
substitute 4 to 3
2tan(x) (1 + 5tan²(x))/(1-tan²(x)) =4tan(x)
2tan(x) (1 + 5tan²(x))=4tan(x)(1-tan²(x))
2tan(x) + 10tan³(x))=4tan(x)-4tan³(x)
14tan³(x)=2tan(x)
14(sin³(x)/cos³(x))=2(sin(x)/cos(x))
14(sin²(x)/cos²(x))=2
14sin²(x)=2cos²(x)
14sin²(x)=2(1-sin²(x))
14sin²(x)+2sin²(x)=2
16sin²(x)=2
sin(x)=√(2/16) or
sin (x)=√2/4 ans.
where x=20.705^o

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