Expressing sin(x) and cos(x) in terms of t = tan(x/2)

In this video, I derive the following identities:

sin(x) = 2*t / (1 + t^2)
cos(x) = (1 - t^2) / (1 + t^2)

These identities are important when it comes to simplifying integrals of rational fractions involving sin(x), cos(x) and tan(x).

The process is simple, but we need to remember the double angle formulas:

sin(2a) = 2*sin(a)*cos(a)
cos(2a) = cos^2(a) - sin^2(a)

We also need to remember that in terms of tan(x):

sin(x) = tan(x) / sqrt( 1 + tan^2(x) )
cos(x) = 1 / sqrt( 1 + tan^2(x) )

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

[DianaGarcia-hcmb]

just exactly what everyone in class is looking for! We couldn't find it in anywhere but here! Thank youuu

[beatbahlek]

I know everybody said it already but thanks a lot! I only found the answer to the problem in one space but nowhere could I find the working to it.
Thank you very much indeed! It finally makes sense!

[kenzzz]

Okay, I know this is a really old video but damn, thanks man! It helped a lot!

[Ravasandani]

thanks for the video! can you then rewrite (1 - t^2 / 1 + t^2) as (1 - t / 1 + t ) ?

[leigh-annamoroso]

Thank you for this wonderful video. You made it very easy to understand.

[housseinsahnun]

i appreciate the video and the effort sir.. i truly used every single information in it.. thank u :)

[norman]

Thank you thank you thank you thank you thank you thank you thank you

[monopole-uwu]

Thanks for the amazing video! Just one question though: since sinx = tanx/secx and sec^2 x = 1+tan^2 x, what is the reason for dropping the negative square root?