Question

Given $a={\sec }^2\alpha ,b={cosec}^2\alpha ,and,c=\frac{1}{1-{\sin }^2\alpha {\cos }^2\alpha }.$ The value of $bc+ca-ab$ is

• A. $a$
• B. $b$
• C. $c$
• D. $-c$

Answer 1

Answer: (C)

$c$

We know that,

${sin}^2\theta +{cos}^2\theta =1$

Given that, $a={\sec }^2\alpha ,b={cosec}^2\alpha ,$

$\Rightarrow$ $\frac{1}{a}={cos}^2\alpha ,\frac{1}{b}={sin}^2\alpha$

$\frac{1}{a}+\frac{1}{b}=1$
and $c=\frac{1}{1-\frac{1}{ab}}=\frac{1}{\frac{1}{a}+\frac{1}{b}-\frac{1}{ab}}=\frac{ab}{a+b-1}$

$\therefore ac+bc-c=ab$,

Thus,
$ac+bc-ab=c$

Hence, option 'C' is correct.