Question

If $\tan x=1\frac{1}{3},$ find the value of: $4{\sin }^{2}x-3{\cos }^{2}x+2$

• A. $\frac{12}{25}$
• B. $2\frac{12}{25}$
• C. $3\frac{12}{25}$
• D. $4\frac{12}{25}$

Answer 1

The correct option is C $3\frac{12}{25}$

$\tan x=1\frac{1}{3}$

$\tan x=\frac{4}{3}$

$\tan x=\frac{P}{B}=\frac{4}{3}$

Using pythagoras Theorem,

$H^2=P^2+B^2$

$H^2=4^2+3^2$

$H=5$

Now, $4{\sin }^{2}x-3{\cos }^{2}x+2$

$=4{\left(\frac{P}{H}\right)}^2-3{\left(\frac{B}{H}\right)}^2+2$

$=4{\left(\frac{4}{5}\right)}^2-3{\left(\frac{3}{5}\right)}^2+2$

$=\frac{64}{25}-\frac{27}{25}+2$

$=\frac{87}{25}$

​$=3\frac{12}{25}$