Question

tan(1/2(cos-1{√5/3}))

Answer 1

In the attachments I have answered this problem.

Half angle formula has been applied in this problem.

See the attachment for detailed solution.

Answer 2

Answer:

Step-by-step explanation:

The given equation is:

$tan(\frac{1}{2}(cos^{-1}{\frac{\sqrt{5}}{{3}})$

Let us assume that $(cos^{-1}{\frac{\sqrt{5}}{{3}})=\theta$

⇒${\frac{\sqrt{5}}{{3}}=cos{\theta}$

Now, $tan(\frac{1}{2}(cos^{-1}{\frac{\sqrt{5}}{{3}})=tan{\frac{\theta}{2}}=x$ (Say)

Also, $cos\theta=\frac{1-tan^2{\frac{\theta}{2}}}{1+tan^2{\frac{\theta}{2}}}}$

${\frac{\sqrt{5}}{3}}=\frac{1-x^2}{1+x^2}$

$\sqrt{5}+\sqrt{5}x^2=3-3x^2$

$(\sqrt{5}+3)x^2=3-\sqrt{5}$

$x^2=\frac{3-\sqrt{5}}{3+\sqrt{5}}$

$x=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}$

which is the required solution.