Question

tan theta - sin theta = 0.25 find theta

Answer 1

I will just start over assuming you know these substations.
\[\tan \theta- \sin \theta = 1/4\]
\[4\sin \theta (\frac{ 1 }{ \cos \theta }-1) = 1.\]
\[4\sin \theta[\frac{ 1- \cos \theta }{ \cos \theta}]=1\]
Squaring.
\[16\sin ^{2} \theta [\frac{ \cos^2 \theta - 2 \cos \theta+1 }{ \cos ^{2}\theta }] = 1\]
\[16 (1-\cos ^{2}\theta)(\cos ^{2}\theta -2\cos \theta + 1) = \cos ^{2}\theta.\]
\[16 \cos ^{4}\theta -2\cos ^{3}\theta +\cos ^{2}\theta +2\cos \theta-1=0\]
Now use any software or take this https://en.wikipedia.org/wiki/Quartic_function#Solving_a_quartic_equation
to solve it for you.

Answer 2

Oops! I forgot to multiply terms by 16.
16()-32()+16()+32()-16=0