Is sin(x)cos(x)+sin^3(x)sec(x)=tan(x) ?

The answer to whether sin(x)cos(x)+sin^3(x)sec(x)=tan(x) is True

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prove sin(x)cos(x)+sin^3(x)sec(x)=tan(x)

prove

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

True

Solution steps

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

Manipulating left side

sin (x) cos (x) + sin^{3} (x) sec (x)

Express with sin, cos

= \frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )}

Factor

\frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )} : \frac{sin ( x ) ( sin ^{2} ( x ) + cos ^{2} ( x ) )}{cos ( x )}

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (x) cot^{2} (x) + sin^{2} (x) = 1

- tan (θ) cos (θ) = sin (- θ)

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

sin^{2} (2 x) = \frac{1}{2} (1 - cos (4 x))

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

Is sin(x)cos(x)+sin^3(x)sec(x)=tan(x) ?
The answer to whether sin(x)cos(x)+sin^3(x)sec(x)=tan(x) is True