Is sin^2(x)sec^2(x)= 1/(cot^2(x)) ?

The answer to whether sin^2(x)sec^2(x)= 1/(cot^2(x)) is True

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prove sin^2(x)sec^2(x)= 1/(cot^2(x))

prove

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

True

Solution steps

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{sin ^{2} ( x )}{( \frac{1}{s e c ( x )} ) ^{2}}

Simplify

\frac{sin ^{2} ( x )}{( \frac{1}{s e c ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sin^{2} (x) sec^{2} (x)

sin^{2} (x) sec^{2} (x)

sin^{2} (x) sec^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (x) = cos (x)

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

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

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

cos (a - b) - cos (a + b) = 2 sin (a) sin (b)

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