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

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

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

prove

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

True

Solution steps

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

cos (2 A + A) = - 3 cos (A) + 4 cos^{3} (A)

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

(sin^{2} (x) + cos^{2} (x))^{3} = 1^{3}

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