3. The value of the expression,$\mathrm{cosec} (75° + \mathrm{\theta }) – \sec (15° – \mathrm{\theta }) – \tan (55° + \mathrm{\theta }) + \cot (35° – \mathrm{\theta })$ is ____.

According to the question,

We have to find the value of the equation,

$\mathrm{cosec}(75°+\mathrm{\theta }) – \sec (15°-\mathrm{\theta }) – \tan (55°+\mathrm{\theta }) + \cot (35°-\mathrm{\theta })$

$=\mathrm{cosec}[90°-(15°-\mathrm{\theta }\left)\right] – \sec (15°-\mathrm{\theta }) – \tan (55°+\mathrm{\theta }) + \cot [90°-(55°+\mathrm{\theta }\left)\right]$

Since, $\mathrm{cosec} (90°- \mathrm{\theta }) = \sec \mathrm{\theta }$

And, $\cot (90°-\mathrm{\theta }) = \tan \mathrm{\theta }$

We get,

$\sec (15°-\mathrm{\theta }) – \sec (15°-\mathrm{\theta }) – \tan (55°+\mathrm{\theta }) + \tan (55°+\mathrm{\theta })$

$=0$