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

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

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prove tan^2(a)+1= 1/(cos^2(a))

prove

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

True

Solution steps

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

We showed that the two sides could take the same form

\Rightarrow True

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

- csc (x) cos (x) = - cot (x)

sin^{2} (y) cos^{2} (y) = \frac{1 - cos ( 4 x )}{8}

cos (θ) tan (θ) sin (θ) = sin^{2} (θ)

cos (\frac{3 x}{5}) = - sin (\frac{3 x}{5} - \frac{π}{2})

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