Write 2 tan theta/2 cos^2 theta/2 in terms of single angle

[adpcane15]

Use identities to write the expression in terms of a single angle Ө.

2tan Ө/2 cos^2 Ө/2

 

[stapel]

What have you tried? How far have you gotten? Where are you stuck?

Please be complete and specific. Thank you.

Eliz.

 

[soroban]

Re: Write 2 tan theta/2 cos^2 theta/2 in terms of single ang

Hello, adpcane15!

This one is quite simple.
. . Are you familiar with any of the identities?

Use identities to write the expression in terms of a single angle \(\displaystyle \theta.\)

\(\displaystyle 2\cdot\tan\left(\frac{\theta}{2}\right)\cdot\cos^2\left(\frac{\theta}{2}\right)\)

We have: \(\displaystyle \L\:2\cdot\frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)}\,\cdot\,\cos^2\left(\frac{\theta}{2}\right) \;=\;2\cdot\sin\left(\frac{\theta}{2}\right)\cdot\cos\left(\frac{\theta}{2}\right) \;=\;sin\theta\)

 

[jonboy]

Soroban you lost me lol.

How is \(\displaystyle \L \;Sin\frac{\theta}{2}\,=\,\frac{Sin\,\frac{\theta}{2}}{cos\,\frac{\theta}{2}}\) ?

 

[stapel]

How is \(\displaystyle \L \;Sin\frac{\theta}{2}\,=\,\frac{Sin\,\frac{\theta}{2}}{cos\,\frac{\theta}{2}}\) ?

I think you overlooked an exponent. The equality in question boils down to:

. . . . .\(\displaystyle \frac{\sin{(x)}}{\cos{(x)}}\, \frac{\left(\cos{(x)}\right)^2}{1}\, =\, \sin{(x)}\, \cos{(x)}\)

The cosine after the first fraction is squared.

Eliz.

 

[jonboy]

Yep I overlooked it. Good read Stapel! Now that I understand everything, I can admire Soroban's work even more!