Question
Evaluate:
$\text{cosec }(65^\circ+\theta)-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\cot(35^\circ+\theta)$
$\text{cosec }(65^\circ+\theta)-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\cot(35^\circ+\theta)$
✓
Answer
$\text{cosec}(65^\circ+\theta)-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\cot(35^\circ+\theta)$
$ = \text{cosec}\big\{90^\circ-(25^\circ-\theta)\big\}-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\cot\big\{90^\circ-(55^\circ-\theta)\big\}$
$ =\sec(25^\circ-\theta)-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\tan(55^\circ-\theta)$
$= 0$
$ = \text{cosec}\big\{90^\circ-(25^\circ-\theta)\big\}-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\cot\big\{90^\circ-(55^\circ-\theta)\big\}$
$ =\sec(25^\circ-\theta)-\sec(25^\circ-\theta)-\tan(55^\circ-\theta)+\tan(55^\circ-\theta)$
$= 0$
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