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

cos (4 π + x) = cos (x)

5 cot^{2} (y) (sec^{2} (y) - 1) = 5

sin (x) cos (y) + sin (y) cos (x) = sin (x + y)

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

\frac{sin ( 7 x ) - sin ( 5 x )}{cos ( 7 x ) + cos ( 5 x )} = tan (x)

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