упростить tan(1+tan^2(x))

Решение

упростить

tan (1 + tan^{2} (x))

tan (sec^{2} (x))

Шаги решения

tan (1 + tan^{2} (x))

Перепишите используя тригонометрические тождества

= \frac{sin ( 1 ) cos ( \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} ) + cos ( 1 ) sin ( \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} )}{cos ( 1 ) cos ( \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} ) - sin ( 1 ) sin ( \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} )}

Испльзуйте основное тригонометрическое тождество:

\frac{sin ( x )}{cos ( x )} = tan (x)

= \frac{sin ( 1 ) cos ( tan ( x ) tan ( x ) ) + cos ( 1 ) sin ( tan ( x ) tan ( x ) )}{cos ( 1 ) cos ( tan ( x ) tan ( x ) ) - sin ( 1 ) sin ( tan ( x ) tan ( x ) )}

sin (1) cos (tan (x) tan (x)) + cos (1) sin (tan (x) tan (x)) = sin (1) cos (tan^{2} (x)) + cos (1) sin (tan^{2} (x))

= \frac{sin ( 1 ) cos ( tan ^{2} ( x ) ) + cos ( 1 ) sin ( tan ^{2} ( x ) )}{cos ( 1 ) cos ( tan ( x ) tan ( x ) ) - sin ( 1 ) sin ( tan ( x ) tan ( x ) )}

cos (1) cos (tan (x) tan (x)) - sin (1) sin (tan (x) tan (x)) = cos (1) cos (tan^{2} (x)) - sin (1) sin (tan^{2} (x))

= \frac{sin ( 1 ) cos ( tan ^{2} ( x ) ) + cos ( 1 ) sin ( tan ^{2} ( x ) )}{cos ( 1 ) cos ( tan ^{2} ( x ) ) - sin ( 1 ) sin ( tan ^{2} ( x ) )}

Используйте тождество суммы углов:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= \frac{sin ( 1 + tan ^{2} ( x ) )}{cos ( 1 ) cos ( tan ^{2} ( x ) ) - sin ( 1 ) sin ( tan ^{2} ( x ) )}

Упростить

cos (1) cos (tan^{2} (x)) - sin (1) sin (tan^{2} (x)) : cos (tan^{2} (x) + 1)

= \frac{sin ( 1 + tan ^{2} ( x ) )}{cos ( 1 + tan ^{2} ( x ) )}

Упростить

1 + tan^{2} (x) : sec^{2} (x)

= \frac{sin ( 1 + tan ^{2} ( x ) )}{cos ( sec ^{2} ( x ) )}

Упростить

sin (1 + tan^{2} (x)) : sin (sec^{2} (x))

= \frac{sin ( sec ^{2} ( x ) )}{cos ( sec ^{2} ( x ) )}

Испльзуйте основное тригонометрическое тождество:

\frac{sin ( x )}{cos ( x )} = tan (x)

= tan (sec^{2} (x))