Question

Evaluate: $\frac{{\cos }^2{20}^o+{\cos }^2{70}^o}{{\sec }^2{50}^o-{\cot }^2{40}^o}+2{\text{cosec}}^2{58}^o-2\cot {58}^o\tan {32}^o-4\tan {13}^o\tan {37}^o\tan {45}^o\tan {53}^o\tan {77}^o$

• A. $2$
• B. $-1$
• C. $-2$
• D. $3$

Answer 1

Answer: (B)

$-1$

$\frac{{\cos }^2{20}^o+{\cos }^2{70}^o}{{\sec }^2{50}^o+{\cot }^2{40}^o}+2{\csc }^2{58}^o-2\cot {58}^o\tan {32}^o-4\tan {13}^o\tan {37}^o\tan {45}^o\tan {53}^o\tan {77}^o$
$=\frac{{\cos }^2{({90}^o-70}^o)+{\cos }^2{70}^o}{{\sec }^2{({90}^o-40}^o)+{\cot }^2{40}^o}$

$+2{\csc }^2{58}^o-2\cot {58}^o\tan ({90}^o-{58}^o)-4\left(\tan {13}^o\tan {77}^o\right)\left(\tan {37}^o\tan {53}^o\right)\tan {45}^o$
$=\frac{{\sin }^2{70}^o+{\cos }^2{70}^o}{co{\sec }^2{40}^o-{\cot }^2{40}^o}+2({\csc }^2{58}^o-{\cot }^2{58}^o)-4\left[\tan ({90}^o-{77}^o)\tan {77}^o\right]\left[\tan ({90}^o-{53}^o)\tan {53}^o\right]\left[1\right]$
$=\frac{1}{1}+2-4$
$=-1$