Is cos(x)(1+tan(x))^2=sec(x)+2sin(x) ?

The answer to whether cos(x)(1+tan(x))^2=sec(x)+2sin(x) is True

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prove cos(x)(1+tan(x))^2=sec(x)+2sin(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Expand

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

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

Rewrite using trig identities

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

Rewrite using trig identities

sec (x) + 2 sin (x)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (θ) = \frac{sec ( θ )}{csc ( θ )}

\frac{sec ( θ )}{csc ^{2} ( θ )} = sec (θ) - cos (θ)

cos (3 β) = cos (β) (cos^{2} (β) - 3 sin^{2} (β))

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

Is cos(x)(1+tan(x))^2=sec(x)+2sin(x) ?
The answer to whether cos(x)(1+tan(x))^2=sec(x)+2sin(x) is True