Connect the curve 6

Algebra Level 4

We plot two continuous curves \[f\left( x \right) =\sin { \left( x+a \right) } \cos { \left( x-a \right) } \\ g\left( x \right) =\sin { \left( x-a \right) } \cos { \left( x+a \right) } \]

And another curve \[h\left( x \right) =\frac { f\left( x \right) }{ g\left( x \right) } \]

It is observed that for some \(a\), such that \(\frac { \pi }{ 2 } <a<\pi \), the curve of \(h\left( x \right) \) is tangent to the curve of \(f\left( x \right) \) at point \(\left( t,h\left( t \right) \right) \). Here, \(0<t<\frac { \pi }{ 2 } \).

If \(t+a\) can be expressed as \[\frac { A }{ B } { \pi }^{ C }+D\] such that \(gcd\left( A,B \right) =1\) and \(A,B,C,D\in Z \) , find the value of \[AB+C+D\]

Hint: Differentiate

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