I'm trying to calculate RA and DEC with nutation correction. The direct calculation in MEEUS Page 151 can't be used for stars close to celestial poles and I thought I'd turn RA/DEC into ecliptical longitude and latitude and then add delta psi to the longitude and transform back to RA/DEC. Unfortunately my calculations are wrong somewhere and I can't figure out where I made a mistake. You can find my spreadsheet here: https://1drv.ms/f/s!AukrJ2RhCvoHh0w8MygI0qajeoXt
The standard methods of applying nutation retain equatorial components of position, and for near-polar objects they avoid the drawbacks of the abbreviated approximation by converting the equatorial components of the position into rectangular components and applying the nutation matrix for the date. In principle this method need not involve approximation.
The topic is explained in principle in chapter 3 of 'Explanatory Supplement to the Astronomical Almanac' (ed. P K Seidelmann, 2nd ed, 1992/2006), where the full nutation matrix is specified on p.115 in terms of classical components of nutation in longitude and obliquity, dpsi and deps.
The calculations to evaluate these components has undergone much development over the years, the latest version has a very long list of >1000 trigonometrical components, but a selection of specifications and an approximation is given in the suite of algorithms and codes available at the website for the IAU SOFA software ('Standards of Fundamental Astronomy'), at [http://www.iausofa.org] , and of course any user can apply further shortening approximations to the series.
The IAU suite has three versions of the calculation of the nutation components in longitude and obliquity, routines iau_NUT80 (older standard), iau_NUT00A (full and very long post-2000 standard calculation), and iau_NUT00B (abridged version of the modern calculation). There is also a routine iau_NUMAT for calculating the nutation matrix.
(An intermediate approximation for the components of nutation, useful for applications where abridgment 2000B is too drastically approximated but 2000A is too long, is given in the US Naval Observatory's 'NOVAS' software suite (Naval Observatory Vector Astrometry software) [http://aa.usno.navy.mil/software/novas/novas_info.php]. The NOVAS suite also offers a more integrated approach to calculating astrometric problems , where IAU-SOFA provides individual short modules to be selected and assembled by the user.)
Lastly, the question doesn't mention applying a precession correction as well, but in case that is wanted (which it very often is, in conjunction with nutation), the references already given above provide the needed resources.