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

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

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prove sin(x)sec^2(x)=(tan(x))/(cos(x))

prove

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

True

Solution steps

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

Manipulating left side

sin (x) sec^{2} (x)

Express with sin, cos

sec^{2} (x) sin (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

(\frac{1}{cos ( x )})^{2} sin (x) : \frac{sin ( x )}{cos ^{2} ( x )}

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

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

(\frac{1}{cos ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sin (2 θ) = 2 cot (θ) sin^{2} (θ)

tan (θ) + 1 = sec (θ)

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

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

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