Is (csc^2(x)-1)(sec^2(x)-1)=1 ?

The answer to whether (csc^2(x)-1)(sec^2(x)-1)=1 is True

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prove (csc^2(x)-1)(sec^2(x)-1)=1

prove

(csc^{2} (x) - 1) (sec^{2} (x) - 1) = 1

True

Solution steps

(csc^{2} (x) - 1) (sec^{2} (x) - 1) = 1

Manipulating left side

(csc^{2} (x) - 1) (sec^{2} (x) - 1)

Rewrite using trig identities

(csc^{2} (x) - 1) (sec^{2} (x) - 1)

Use the Pythagorean identity:

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

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

= (csc^{2} (x) - 1) tan^{2} (x)

Use the Pythagorean identity:

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

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= \frac{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

cos (x) + sin (x) \cdot tan (x) = sec (x)

cot (x) + 1 = csc (x) (cos (x) + sin (x))

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

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

\frac{7 cot ( x )}{sec ( x )} = 7 csc (x) - 7 sin (x)

Is (csc^2(x)-1)(sec^2(x)-1)=1 ?
The answer to whether (csc^2(x)-1)(sec^2(x)-1)=1 is True