Question

Prove:

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

Answer 1

Solution :- given

Taking LHS [we know $1+ta{n}^2\theta -se{c}^2\theta ]$

$\Rightarrow se{c}^{4}A(1-si{n}^{4}A)-2ta{n}^{2}A$

$\Rightarrow \frac{1}{co{s}^{4}A}(1-si{n}^{4}A)-2ta{n}^{2}A$

$=se{c}^{4}A-ta{n}^{4}A-2ta{n}^{2}A$

$\Rightarrow (1+ta{n}^{2}A{)}^2-ta{n}^{4}A-2ta{n}^{2}A$

$\Rightarrow 1+2ta{n}^{2}A+ta{n}^{4}A-ta{n}^{4}A-2ta{n}^{2}A$

$\Rightarrow 1$ RHS proved