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

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

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

prove

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

True

Solution steps

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

Manipulating right side

sec (x)

Express with sin, cos

sec (x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

\frac{1}{cos ( x )}

Use the Pythagorean identity:

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

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

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

Expand

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

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

Apply the fraction rule:

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

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

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

Cancel

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

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

Cancel the common factor:

cos (x)

= cos (x)

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{1}{csc ( x ) tan ( x )} = cos (x)

\frac{csc ( x )}{cot ( x )} = csc (x) sec (x)

2 cos (a + b) sin (b) = sin (a + 2 b) - sin (a)

\frac{1}{1 + sin ^{2} ( a )} + \frac{1}{1 + csc ^{2} ( a )} = 1

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