What is tan(1+tan^2(x)) ?

The solution to tan(1+tan^2(x)) is tan(sec^2(x))

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simplify tan(1+tan^2(x))

simplify

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

tan (sec^{2} (x))

Solution steps

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

Rewrite using trig identities

= \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 )} )}

Use the basic trigonometric identity:

\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 ) )}

Use the Angle Sum identity:

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 ) )}

Simplify

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 ) )}

Simplify

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

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

Simplify

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

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

Use the basic trigonometric identity:

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

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

sin (arccos (4 x))

cos (θ - 54 0^{\circ})

\frac{- cos ( x )}{cos ^{2} ( x )}

sin (arccos (\frac{12}{x}))

tan (arcsec (x))