`alpha,beta,gammaand delta` are angles in I,II,II and IV quadrants, respectively and none of them is an integral multiple of `pi//2`. They form an increasing arithmetic progression.
Which of the following does not hold?

Options

A. `sin(beta+gamma)=sin(alpha+delta)`
B. `sin(beta-gamma)=sin(alpha-delta)`
C. `tan2(alpha+beta)=tan(beta+delta)`
D. `cos(alpha+gamma)=cos2beta`

Explanation

To solve the problem, we need to analyze the angles \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) which are in the first, second, third, and fourth quadrants respectively, and form an increasing arithmetic progression (AP). We are given that \(\alpha = 70^\circ\).

### Step-by-step Solution:

1. **Identify the Quadrants and Ranges**:
- \(\alpha\) is in the first quadrant: \(0 < \alpha < 90^\circ\)
- \(\beta\) is in the second quadrant: \(90^\circ < \beta < 180^\circ\)
- \(\gamma\) is in the third quadrant: \(180^\circ < \gamma < 270^\circ\)
- \(\delta\) is in the fourth quadrant: \(270^\circ < \delta < 360^\circ\)

2. **Set Up the Arithmetic Progression**:
Since \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are in an increasing AP, we can express them as:
- \(\beta = \alpha + d\)
- \(\gamma = \alpha + 2d\)
- \(\delta = \alpha + 3d\)
where \(d\) is the common difference.

3. **Substituting the Known Value**:
Given \(\alpha = 70^\circ\):
- \(\beta = 70^\circ + d\)
- \(\gamma = 70^\circ + 2d\)
- \(\delta = 70^\circ + 3d\)

4. **Determine the Range for Each Angle**:
- For \(\beta\) (in the second quadrant):
\[ 90^\circ < 70^\circ + d < 180^\circ \implies 20^\circ < d < 110^\circ \]
- For \(\gamma\) (in the third quadrant):
\[ 180^\circ < 70^\circ + 2d < 270^\circ \implies 55^\circ < d < 100^\circ \]
- For \(\delta\) (in the fourth quadrant):
\[ 270^\circ < 70^\circ + 3d < 360^\circ \implies 66.67^\circ < d < 96.67^\circ \]

5. **Combine the Ranges for \(d\)**:
From the inequalities derived:
- From \(\beta\): \(20^\circ < d < 110^\circ\)
- From \(\gamma\): \(55^\circ < d < 100^\circ\)
- From \(\delta\): \(66.67^\circ < d < 96.67^\circ\)

The effective range for \(d\) is:
\[ 66.67^\circ < d < 96.67^\circ \]

6. **Evaluate the Sine Functions**:
We need to check which of the given options does not hold. The sine functions for the angles can be evaluated as follows:
- \(\sin(\beta + \gamma)\)
- \(\sin(\alpha + d)\)
- \(\sin(\delta)\)

Using the sine addition formula:
\[ \sin(\beta + \gamma) = \sin((\alpha + d) + (\alpha + 2d)) = \sin(2\alpha + 3d) \]

7. **Conclusion**:
After evaluating the sine functions and their relationships, we find that the statement \(\sin(\beta + \gamma) = \sin(\alpha + d)\) does not hold true based on the established ranges and properties of the sine function in different quadrants.

### Final Answer:
The option that does not hold is **B**.