Question

$({\cos }^2\theta -1)({\cot }^2\theta +1)+1=$

• A. $1$
• B. $-1$
• C. $2$
• D. $0$

Answer 1

Answer: (D)

$0$

The value of $({\cos }^2\theta -1)({\cot }^2\theta +1)+1$ is

$={\cos }^2\theta .{\cot }^2\theta +{\cos }^2\theta -{\cot }^2\theta -1+1$

$=\frac{{\cos }^4\theta }{{\sin }^2\theta }+{\cos }^2\theta -\frac{{\cos }^2\theta }{{\sin }^2\theta }$

$=\frac{{\cos }^4\theta +{\cos }^2\theta .{\sin }^2\theta -{\cos }^2\theta }{{\sin }^2\theta }$

$=\frac{{\cos }^2\theta ({\cos }^2\theta +{\sin }^2\theta -1)}{{\sin }^2\theta }$

$=\frac{{\cos }^2\theta (1-1)}{{\sin }^2\theta }$ .... $[$ Since ${\sin }^2\theta +{\cos }^2\theta =1]$

$=0$

Hence, option D is correct.