Question
Prove that:
$\text{cosec}(65^\circ+\theta)-\sec(25^\circ-\theta)\\-\tan(55^\circ-\theta)+\cot(35^\circ+\theta)=0$
$\text{cosec}(65^\circ+\theta)-\sec(25^\circ-\theta)\\-\tan(55^\circ-\theta)+\cot(35^\circ+\theta)=0$
✓
Answer
$\text{L.H.S.}=\text{cosec}(65^\circ+\theta)-\sec(25^\circ-\theta)\\ \ \ -\tan(55^\circ-\theta)+\cot(35^\circ+\theta)$
$=\text{cosec}\big\{90^\circ-(25^\circ-\theta)\big\}-\sec(25^\circ-\theta)\\ \ \ -\tan(55^\circ-\theta)+\cot\big\{90^\circ-(55^\circ-\theta)\big\}$
$=\sec(25^\circ+\theta)-\sec(25^\circ-\theta)\\-\tan(55^\circ-\theta)+\tan(55^\circ-\theta)$
$=0$
$=\text{R.H.S.}$
$=\text{cosec}\big\{90^\circ-(25^\circ-\theta)\big\}-\sec(25^\circ-\theta)\\ \ \ -\tan(55^\circ-\theta)+\cot\big\{90^\circ-(55^\circ-\theta)\big\}$
$=\sec(25^\circ+\theta)-\sec(25^\circ-\theta)\\-\tan(55^\circ-\theta)+\tan(55^\circ-\theta)$
$=0$
$=\text{R.H.S.}$
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