Is 1/(cos^2(θ))=tan^2(θ)+1 ?

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

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prove 1/(cos^2(θ))=tan^2(θ)+1

prove

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

True

Solution steps

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

Manipulating right side

tan^{2} (θ) + 1

Express with sin, cos

1 + tan^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

1 + (\frac{sin ( θ )}{cos ( θ )})^{2}

Apply exponent rule:

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

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

Convert element to fraction:

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

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

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} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )}

Multiply:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= \frac{1}{cos ^{2} ( θ )}

= \frac{1}{cos ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

(sin (θ) - 1) (tan (θ) + sec (θ)) = - cos (θ)

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

sin (b) cos^{2} (b) - sin (b) = - sin^{3} (b)

\frac{1}{2} (cot (θ) - tan (θ)) = cot (2 θ)

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

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