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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions5 Marks
Question
If $\text{x}=\sec\phi-\tan\phi$ and $\text{y}=\text{cosec}\phi+\cot\phi$ then show that $\text{xy}+\text{x}-\text{y}+1=0$
[Hint: Find xy + 1 and then show that x - y = -(xy + 1)]
✓

Answer

$\text{x}=\sec\phi-\tan\phi\Rightarrow\text{x}=\frac{1-\sin\phi}{\cos\phi}$
$\text{y}=\text{cosec}\phi+\cot\phi\Rightarrow\text{y}=\frac{1+\cos\phi}{\sin\phi}$
$\Rightarrow\text{xy + x}-\text{y}=\frac{1-\sin\phi1+\cos\phi}{\cos\phi\sin\phi}+\frac{1-\sin\phi}{\cos\phi}-\frac{1+\cos\phi}{\sin\phi}$
$=\frac{(1-\sin\phi)(1+\cos\phi)+(1-\sin\phi)\sin\phi-\cos\phi(1+\cos\phi)}{\sin\phi\cos\phi}$
$=\frac{1-\sin\phi+\cos\phi-\sin\phi\cos\phi+\sin\phi-\sin^2\phi-\cos\phi-\cos^2\phi}{\sin\phi\cos\phi}$
$=\frac{1-\sin\phi\cos\phi-(\sin^2\phi+\cos^2\phi)}{\sin\phi\cos\phi}=-1$
$\therefore\text{xy + x}-\text{y}-1=0$

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