Is tan^2(A)+1=(sec(A))/(cos(A)) ?

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

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prove tan^2(A)+1=(sec(A))/(cos(A))

prove

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

True

Solution steps

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

Manipulating left side

tan^{2} (A) + 1

Express with sin, cos

1 + tan^{2} (A)

Use the basic trigonometric identity:

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

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

Simplify

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

1 + (\frac{sin ( A )}{cos ( A )})^{2}

Apply exponent rule:

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos^{2} (A) = cos^{2} (A)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Manipulating right side

\frac{sec ( A )}{cos ( A )}

Express with sin, cos

\frac{sec ( A )}{cos ( A )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{c o s ( A )}}{cos ( A )}

Simplify

\frac{\frac{1}{c o s ( A )}}{cos ( A )} : \frac{1}{cos ^{2} ( A )}

\frac{\frac{1}{c o s ( A )}}{cos ( A )}

Apply the fraction rule:

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

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

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

cos (A) cos (A)

Apply exponent rule:

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

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

= cos^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= cos^{2} (A)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

tan^{2} (x) - sin^{2} (x) = tan (x) sin (x)

sec (- x) = - sec (x)

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

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

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