Question 53

What is the value of $$\frac{[1 +2 \cot^2(90 - x) - 2\cosec(90 - x) \cot(90 - x)]}{[\cosec(90 - x) - \cot(90 - x)]}$$?

Solution

$$\frac{[1+2\cot^2(90-x)-2\operatorname{cosec}(90-x)\cot(90-x)]}{[\operatorname{cosec}(90-x)-\cot(90-x)]}$$

$$=\frac{[1+2\tan^2(x)-2\sec(x)\tan(x)]}{[\sec(x)-\tan(x)]}$$

$$=\frac{[\sec^2x-\tan^2x+2\tan^2(x)-2\sec(x)\tan(x)]}{[\sec(x)-\tan(x)]}$$

$$=\frac{[\sec\left(x\right)-\tan\left(x\right)]^2}{[\sec(x)-\tan(x)]}\ .$$

$$=[\sec\left(x\right)-\tan\left(x\right)]\ .$$

D is correct choice.

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