Is sec(x)-cos(x)=sin^2(x)sec(x) ?

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

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prove sec(x)-cos(x)=sin^2(x)sec(x)

prove

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

True

Solution steps

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

Manipulating left side

sec (x) - cos (x)

Express with sin, cos

- cos (x) + sec (x)

Use the basic trigonometric identity:

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

= - cos (x) + \frac{1}{cos ( x )}

Simplify

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

- cos (x) + \frac{1}{cos ( x )}

Convert element to fraction:

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

= - \frac{cos ( x ) cos ( x )}{cos ( x )} + \frac{1}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ( x ) cos ( x ) + 1}{cos ( x )}

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

- cos (x) cos (x) + 1

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

csc^{4} (x) - cot^{4} (x) = 2 csc^{2} (x) + 1

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

2 sin^{2} (x) = 1

\frac{1 - cos ^{2} ( θ )}{1 - sin ^{2} ( θ )} = tan^{2} (θ)

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

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