I need to solve for x in
sin(x)=3cos(x)
So I did the following:
sin(x)=3cos(x)sin2(x)=9cos2(x)(squaring both sides)0=9cos2(x)−sin2(x)(subtracting sin2(x))
My question is: Am I allowed to use the identity
cos(2x)=cos2(x)−sin2(x)
in the equation to make it
0=9cos2(x)−sin2(x)→0=9cos(2x)
or is that the case that
9cos(2x)≠9cos2(x)−sin2(x)
because the 9 is multiplying the cos only, so that I'm not allowed to use this identity?
Answer
Hint: once squared, you can use the identity:
sin2(x)+cos2(x)=1
To obtain
sin2(x)=9cos2(x)=9⋅(1−sin2(x)).
From here you get
sin2(x)=9/10
And you can finish the calculation...
There is no need to use the double-angle identity
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