Find the sum of inverse functions

Question

$y=\sin^{-1}(\sin 8)-\tan^{-1}(\tan 10)+\cos^{-1}(\cos 12)- \sec^{-1}(\sec 9)+\cot^{-1}(\cot 6)-\csc^{-1}(\csc 7)$.

If $y$ simplifies to $y=aπ+b$, then find $a-b$.

My answer is zero.

But the answer is 53. Where have I gone wrong?

Answer 1

Hint:

Using principal values of inverse trigonometric functions

$$2\pi<8<3\pi\implies\sin^{-1}(\sin8)=3\pi-8$$

$$3\pi<10<4\pi\implies\tan^{-1}(\tan10)=10-3\pi$$