Question

demonstre as seguintes identidades trigonométricas:
a) (1+senx)(1-senx) = cos²x
b) tg²x•cossec²x = 1+tg²x
c) cossec²x•tgx = cotgx•sec²x

Answer 1

a)
$(1+senx)(1-senx) = cos^2x$
$1-senx+senx-sen^2x = cos^2x$
$1-sen^2x=cos^2x$
$\boxed {\boxed {Resposta: 1 = cos^2x+sen^2x}}$
Primeira identidade trigonométrica

b)
$tg^2x*cossec^2x = 1+tg^2x$
$\boxed { \frac{sen^2x}{cos^2x} * \frac{1}{sen^2x} = 1+ \frac{sen^2x}{cos^2x} }$
$\boxed {\frac{1}{cos^2x} = 1+ \frac{sen^2x}{cos^2x} }$
$\boxed {\frac{1}{cos^2x} -\frac{sen^2x}{cos^2x}= 1} }$
$\boxed {\frac{1-sen^2x}{cos^2x}= 1} }$
$\boxed {1-sen^2x= 1*cos^2x} }$
$\boxed {1-sen^2x= cos^2x} }$
$\boxed {\boxed {Resposta:cos^2x+sen^2x=1} }$
Primeira identidade trigonométrica

c)
$cossec^2x*tgx = cotgx*sec^2x$
$\boxed {\frac{1}{sen^2x}* \frac{senx}{cosx} = \frac{cosx}{senx} * \frac{1}{cos^2x} }$
$\boxed {\frac{1}{senx*cosx} =\frac{1}{senx*cosx}}$
$\boxed {\boxed {Resposta : 1=1}}$