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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= \frac{csc ^{2} ( x )}{sec ^{2} ( x )}

= \frac{csc ^{2} ( x )}{sec ^{2} ( x )}

Express with sin, cos

\frac{csc ^{2} ( x )}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Divide fractions:

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

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

Refine

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

cot (x) cot (x)

Simplify

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cot (x) cot (x) = cot^{1 + 1} (x)

= cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (x) - cot (x) = - 2 cot (2 x)

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

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

cos^{4} (x) - sin^{4} (x) = 2 cos^{2} (x) - 1

tan (x) (cot (x) + tan (x)) = sec^{2} (x)

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