Is (sec^2(x))/(csc^2(x))=tan^2(x) ?

The answer to whether (sec^2(x))/(csc^2(x))=tan^2(x) is True

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prove (sec^2(x))/(csc^2(x))=tan^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Divide fractions:

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

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

Refine

= \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

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

tan (x) tan (x)

Simplify

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

tan (x) tan (x)

Apply exponent rule:

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

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

= tan^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (A + B) + tan (A - B) = \frac{( 2 sin ( 2 A ) )}{( cos ( 2 A ) + cos ( 2 B ) )}

cos (3 θ) = 4 cos^{3} (θ) - cos (θ)

\frac{1 + csc ( b )}{cot ( b ) + cos ( b )} = sec (b)

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

sin (4 a) = 2 sin (2 a) cos (2 a)

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