Trigonometric Equations

Given the equations:
‌sin‌θ1+sin‌θ2=−‌

21

65

‌cos‌θ1+cos‌θ2=−‌

27

65

By squaring both equations and adding them, we have:
cos2θ1+cos2θ2+sin‌2θ1+sin‌2θ2+2[cos(θ1−θ2)]
This equates to:
=(‌

21

65

)2+(‌

27

65

)2
Calculating further, we get:
2+2‌cos(θ1−θ2)=‌

441+729

652

Simplifying, we have:
2[2cos2(‌

θ1−θ2

2

)]=‌

1170

652

Therefore:
cos(‌

θ1−θ2

2

)=‌

−3

√130

Given that ‌

θ1−θ2

2

lies in the second quadrant, the result confirms cos will be negative in this quadrant. Hence:
cos(‌

θ1−θ2

2

)=−‌

3

√130