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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{1}{1 + tan ^{2} ( x )}

Express with sin, cos

\frac{1}{1 + tan ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

Simplify

\frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}} : \frac{cos ^{2} ( x )}{cos ^{2} ( x ) + sin ^{2} ( x )}

\frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

Apply exponent rule:

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

= \frac{1}{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

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

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

= \frac{1}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ^{2} ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Apply rule

\frac{a}{1} = a

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sec^{4} (x) = \frac{1}{cos ^{4} ( x )}

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

2 cos^{3} (x) - cos (x) = \frac{cos ^{2} ( x ) - sin ^{2} ( x )}{sec ( x )}

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

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