If \cos y = \frac{1}{10}cosy= 10 1 ​ , then what is the positive value of \tan\tfrac{1}{2}ytan 2 1 ​ y, in simplest radical form with a rational denominator

Answer 1

Because of the way this question was written, I can't be certain of its intended meaning. Based on the context of the question, I'm pretty sure that the following text provides the answer you were looking for.

I'm assuming the question is:

If cos(y) = 1/10, then what is the positive value of tan((1/2)y) in simplest radical form?

Solution.

We will use the following half-angle identity:

tan((1/2)y) = ± [ √(1− cos(y)) / √(1+ cos(y)) ]

Since we're given the fact that cos(y) = 1/10, the previous equation can be rewritten as

tan((1/2)y) = ± [ √(1− 1/10) / √(1+ 1/10) ] .

We then compute the right hand side of the previous equation:

± [ √(1− 1/10) / √(1+ 1/10) ] = ± [ √(9/10) / √(11/10) ]

= ± [ √(9) / √(11) ] <---------- cancel out the tens from the previous expression,

= ± [ √(99) / 11 ] <---------- multiply previous expression by √(11) / √(11),

= ± [ 3√(11) / 11 ] <---------- since 99 = 9•11, we have √(99) = √(9•11) = 3√(11).

Since the question only asks for the positive value of tan((1/2)y), our answer is 3√(11) / 11.