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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Apply exponent rule:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Apply rule

\frac{a}{1} = a

= cos (x) sin (x)

= cos (x) sin (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sin (u) (csc (u) - sin (u)) = cos^{2} (u)

cos^{2} (θ) - sin^{2} (θ) = 1

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

\frac{1 - tan ^{4} ( θ )}{sec ^{2} ( θ )} = 1 - tan^{2} (θ)

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