Question

${\text{cosec}}^{2}A{\cot }^{2}A-{\sec }^{2}A{\tan }^{2}A-({\cot }^{2}A-{\tan }^{2}A)({\sec }^{2}A+{\text{cosec}}^{2}A-1)$ is equal to

• A. $0$
• B. $1$
• C. ${\sec }^{2}A$
• D. ${\text{cosec}}^{2}A$

Answer 1

Answer: (A)

$0$

${\text{cosec}}^{2}A{\cot }^{2}A-{\sec }^{2}A{\tan }^{2}A-({\cot }^{2}A-{\tan }^{2}A)({\sec }^{2}A+{\text{cosec}}^{2}A-1)$
$={\text{cosec}}^{2}A{\cot }^{2}A-{\sec }^{2}A{\tan }^{2}A-{\cot }^{2}A{\sec }^{2}A-{\cot }^{2}A{\text{cosec}}^{2}A+{\cot }^{2}A$

$+{\tan }^{2}A{\sec }^{2}A+{\tan }^{2}A{\text{cosec}}^{2}A-{\tan }^{2}A$

$=-{\cot }^{2}A{\sec }^{2}A+{\cot }^{2}A+{\tan }^{2}A{\text{cosec}}^{2}A-{\tan }^{2}A$

$=-{\cot }^{2}A({\sec }^{2}A-1)+{\tan }^{2}A({\text{cosec}}^{2}A-1)$
$=-{\cot }^{2}A{\tan }^{2}A+{\tan }^{2}A{\cot }^{2}A$

$=-1+1$
$=0$