<s>[INST] If $\tan (\pi \cos \theta) =\cot (\pi \sin \theta) $ then find the value of $\cos\left (\theta -\frac{\pi}{4}\right)$ If $\tan (\pi \cos \theta) =\cot (\pi \sin \theta) $ then find the value of $\cos \left(\theta -\frac{\pi}{4}\right).$
I could not get any idea to solve.  However I tried by using $\theta =0^\circ $. But could not get the answer. 
 [/INST] Rearrange into sines and cosines. Form a cosine addition identity.
\begin{align}
\tan(\pi\cos t)\tan(\pi\sin t) &= 1
\\
\sin(\pi\cos t)\sin(\pi\sin t) &= \cos(\pi\cos t)\cos(\pi\sin t)
\\
0 &= \cos(\pi\cos t)\cos(\pi\sin t) - \sin(\pi\cos t)\sin(\pi\sin t)
\\
0 &= \cos(\pi\cos t + \pi\sin t)
\\
0 &= \cos(\pi[\cos t + \sin t])
\end{align}
The argument of the cosine in the previous line must be an odd integer multiple of $\pi/2$, so we get
\begin{align}
\cos t + \sin t &= \frac{2n+1}{2} 
\end{align}
where $n\in\mathbb{Z}$ (EDIT: More accurately, $n \in \{-1,0\}$ to be in the range of $\cos t + \sin t$). Use the cosine subtraction formula to achieve the desired result.
\begin{align}
\frac{1}{\sqrt2}\cos t + \frac{1}{\sqrt2}\sin t &= \frac{2n+1}{2\sqrt{2}}
\\
\cos (t)\cos(\pi/4) + \sin (t)\sin(\pi/4) &= \frac{2n+1}{2\sqrt{2}}
\\
\\
\cos(t -\pi/4) &= \frac{2n+1}{2\sqrt{2}}
\end{align}
EDIT: Following lab bhattacharjee's comment, the range of cosine restricts the possible solutions to 
$$\cos(t -\pi/4) = \frac{\pm1}{2\sqrt{2}}.$$
It doesn't appear like the answer is unique to me. 
</s>