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

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

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prove cot^2(x)sin^2(x)csc^2(x)tan^2(x)=1

prove

cot^{2} (x) sin^{2} (x) csc^{2} (x) tan^{2} (x) = 1

True

Solution steps

cot^{2} (x) sin^{2} (x) csc^{2} (x) tan^{2} (x) = 1

Manipulating left side

cot^{2} (x) sin^{2} (x) csc^{2} (x) tan^{2} (x)

Express with sin, cos

cot^{2} (x) csc^{2} (x) sin^{2} (x) tan^{2} (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

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

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

Apply exponent rule:

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

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

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

Cancel the common factor:

sin^{2} (x)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

\frac{sin ( θ ) sec ( θ )}{tan ( θ )} = 1

\frac{( sin ( θ ) + sin ( 3 θ ) )}{( cos ( θ ) + cos ( 3 θ ) )} = tan (2 θ)

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