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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (θ) cos (θ) + sin^{3} (θ) sec (θ)

Express with sin, cos

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

Factor

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

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

Rewrite using trig identities

= \frac{sin ( θ )}{cos ( θ )}

Use the basic trigonometric identity:

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

= tan (θ)

We showed that the two sides could take the same form

\Rightarrow True

tan (A) sin (A) = sec (A) - cos (A)

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

sin (x) sec (x) = 1

sec^{2} (x) sin^{2} (x) = sec^{2} (x) - 1

tan (a - b) = sin (a - b) sec (a - b)

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