Answer
True.
Work Step by Step
We know that $sin$, $csc$, $tan$ are odd functions, which means $f(-\theta)=-f(\theta).$ Hence $sin(-\theta)=-sin(\theta).$
We also know that $cos$, $sec$, are even functions, which means $f(-\theta)=f(\theta).$ Hence $cos(-\theta)=cos(\theta).$
Therefore $sin(-\theta)+cos(-\theta)=-sin(\theta)+=cos(\theta).$ Hence it is true.