Is sec(θ)tan(θ)=sin(θ)sec^2(θ) ?

The answer to whether sec(θ)tan(θ)=sin(θ)sec^2(θ) is True

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prove sec(θ)tan(θ)=sin(θ)sec^2(θ)

prove

sec (θ) tan (θ) = sin (θ) sec^{2} (θ)

True

Solution steps

sec (θ) tan (θ) = sin (θ) sec^{2} (θ)

Manipulating left side

sec (θ) tan (θ)

Express with sin, cos

sec (θ) tan (θ)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( θ )} tan (θ)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

Simplify

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

\frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot sin ( θ )}{cos ( θ ) cos ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{sin ( θ )}{cos ^{2} ( θ )}

= \frac{sin ( θ )}{cos ^{2} ( θ )}

= \frac{sin ( θ )}{cos ^{2} ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{sin ( θ )}{( \frac{1}{s e c ( θ )} ) ^{2}}

Simplify

\frac{sin ( θ )}{( \frac{1}{s e c ( θ )} ) ^{2}}

(\frac{1}{sec ( θ )})^{2} = \frac{1}{sec ^{2} ( θ )}

(\frac{1}{sec ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sec ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( θ )}

= \frac{sin ( θ )}{\frac{1}{s e c ^{2} ( θ )}}

Apply the fraction rule:

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

= \frac{sin ( θ ) sec ^{2} ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec^{2} (θ) sin (θ)

sec^{2} (θ) sin (θ)

sec^{2} (θ) sin (θ)

= sin (θ) sec^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

(sec (u) - tan (u)) (csc (u) + 1) = cot (u)

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

tan (θ + \frac{π}{4}) = \frac{1 + tan ( θ )}{1 - tan ( θ )}

tan (θ) = \frac{1}{cot ( θ )}

cos (x) tan (x) sec (- x) = sin (x) - 1

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