Question

${\text{sec}}^{4}A(1-{\text{sin}}^{4}A)-2{\text{tan}}^{2}A=$ ___

• A. $\text{tan A}$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

The correct option is B

$1$

${\text{sec}}^{4}A(1-{\text{sin}}^{4}A)-2{\text{tan}}^{2}A$

$=\frac{1}{co{s}^{4}A}(1-{\text{sin}}^{2}A)(1+{\text{sin}}^{2}A)-2{\text{tan}}^{2}A$

$=\frac{{\text{cos}}^{2}A(1+{\text{sin}}^{2}A)}{{\text{cos}}^{4}A}-2{\text{tan}}^{2}A$

$=\frac{1+{\text{sin}}^{2}A}{{\text{cos}}^{2}A}-2{\text{tan}}^{2}A$

$=\frac{1}{co{s}^{2}A}+\frac{si{n}^{2}A}{co{s}^{2}A}-2{\text{tan}}^{2}A$

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

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

$=1(\because 1+ta{n}^{2}A=se{c}^{2}A)$