Thread: Trig - writing an equivalent expression

I'm supposed to write equivalent expressions for these two expressions:

cot(sin^-1(3x))

and

sin(2sin^-1(2x))

I don't even know where to begin with these, I'm sure there's an identity I should be using to write the new expressions...but I can't figure out which one.

Originally Posted by maryanna91

I'm supposed to write equivalent expressions for these two expressions:

cot(sin^-1(3x))

and

sin(2sin^-1(2x))

I don't even know where to begin with these, I'm sure there's an identity I should be using to write the new expressions...but I can't figure out which one.

1) $\displaystyle \cot \left( {{{\sin }^{ - 1}}3x} \right) = \frac{{\cos \left( {{{\sin }^{ - 1}}3x} \right)}}
{{\sin \left( {{{\sin }^{ - 1}}3x} \right)}} = \frac{{\sqrt {1 - {{\left( {3x} \right)}^2}} }}
{{3x}} = \frac{{\sqrt {1 - 9\,{x^2}} }}
{{3x}}.$

2) $\displaystyle \sin \left( {2{{\sin }^{ - 1}}2x} \right) = 2\sin \left( {{{\sin }^{ - 1}}2x} \right)\cos \left( {{{\sin }^{ - 1}}2x} \right) = 2 \cdot 2x\,\sqrt {1 - {{\left( {2x} \right)}^2}} = 4x\,\sqrt {1 - 4\,{x^2}} .$

thanks a lot!