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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Apply the fraction rule:

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

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

\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}}