Is sec^2(a)csc^2(a)=sec^2(a)+csc^2(a) ?

The answer to whether sec^2(a)csc^2(a)=sec^2(a)+csc^2(a) is True

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prove sec^2(a)csc^2(a)=sec^2(a)+csc^2(a)

Solution

prove

sec^{2} (a) csc^{2} (a) = sec^{2} (a) + csc^{2} (a)

Solution

True

Solution steps

sec^{2} (a) csc^{2} (a) = sec^{2} (a) + csc^{2} (a)

Manipulating right side

sec^{2} (a) + csc^{2} (a)

Express with sin, cos

csc^{2} (a) + sec^{2} (a)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

(\frac{1}{sin ( a )})^{2} + (\frac{1}{cos ( a )})^{2} : \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{sin ^{2} ( a ) cos ^{2} ( a )}

(\frac{1}{sin ( a )})^{2} + (\frac{1}{cos ( a )})^{2}

(\frac{1}{sin ( a )})^{2} = \frac{1}{sin ^{2} ( a )}

(\frac{1}{sin ( a )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( a )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( a )}

(\frac{1}{cos ( a )})^{2} = \frac{1}{cos ^{2} ( a )}

(\frac{1}{cos ( a )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( a )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( a )}

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

Least Common Multiplier of

sin^{2} (a), cos^{2} (a) : sin^{2} (a) cos^{2} (a)

sin^{2} (a), cos^{2} (a)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (a)

cos^{2} (a)

= sin^{2} (a) cos^{2} (a)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin^{2} (a) cos^{2} (a)

For

\frac{1}{sin ^{2} ( a )} :

multiply the denominator and numerator by

cos^{2} (a)

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

For

\frac{1}{cos ^{2} ( a )} :

multiply the denominator and numerator by

sin^{2} (a)

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

= \frac{cos ^{2} ( a )}{sin ^{2} ( a ) cos ^{2} ( a )} + \frac{sin ^{2} ( a )}{sin ^{2} ( a ) cos ^{2} ( a )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{sin ^{2} ( a ) cos ^{2} ( a )}

= \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{sin ^{2} ( a ) cos ^{2} ( a )}

= \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a ) sin ^{2} ( a )}

Rewrite using trig identities

\frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a ) sin ^{2} ( a )}

Use the Pythagorean identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{cos ^{2} ( a ) ( \frac{1}{c s c ( a )} ) ^{2}}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

(\frac{1}{csc ( a )})^{2} = \frac{1}{csc ^{2} ( a )}

(\frac{1}{csc ( a )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{csc ^{2} ( a )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( a )}

= \frac{1}{\frac{1}{s e c ^{2} ( a )} \cdot \frac{1}{c s c ^{2} ( a )}}

Multiply

\frac{1}{sec ^{2} ( a )} \cdot \frac{1}{csc ^{2} ( a )} : \frac{1}{sec ^{2} ( a ) csc ^{2} ( a )}

\frac{1}{sec ^{2} ( a )} \cdot \frac{1}{csc ^{2} ( a )}

Multiply fractions:

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

= \frac{1 \cdot 1}{sec ^{2} ( a ) csc ^{2} ( a )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{sec ^{2} ( a ) csc ^{2} ( a )}

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ^{2} ( a ) csc ^{2} ( a )}{1}

Apply rule

\frac{a}{1} = a

= sec^{2} (a) csc^{2} (a)

sec^{2} (a) csc^{2} (a)

sec^{2} (a) csc^{2} (a)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is sec^2(a)csc^2(a)=sec^2(a)+csc^2(a) ?
The answer to whether sec^2(a)csc^2(a)=sec^2(a)+csc^2(a) is True