MCQ
Choose the correct answer from the given four options. The value of the expression $[\text{cosec}(75^\circ+\theta)-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\cot(35^\circ-\theta)]$ is :
- A$–1$
- ✓$0$
- C$1$
- D$\frac{3}{2}$
✓
Answer
Correct option: B.
$0$
Given, expression $=\text{cosec}(75^\circ+\theta)-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\cot(35^\circ-\theta)$
$=\text{cosec}[90^\circ-(15^\circ-\theta)]-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\cot(90^\circ-(55^\circ+\theta))$
$=\sec(15^\circ-\theta)-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\tan(55^\circ+\theta)$
$[\therefore\text{cosec}(90^\circ-\theta)=\sec\theta$ and $\cot(90^\circ-\theta)=\tan\theta]$
$=0$
Hence, the required value of the given expression is $0$.
$=\text{cosec}[90^\circ-(15^\circ-\theta)]-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\cot(90^\circ-(55^\circ+\theta))$
$=\sec(15^\circ-\theta)-\sec(15^\circ-\theta)-\tan(55^\circ+\theta)+\tan(55^\circ+\theta)$
$[\therefore\text{cosec}(90^\circ-\theta)=\sec\theta$ and $\cot(90^\circ-\theta)=\tan\theta]$
$=0$
Hence, the required value of the given expression is $0$.
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