This question requires you to fill in the blanks. Some steps will refer to previous answers you have typed, so it is necessary to "submit" after filling in each blank. The quickest way to do this is hit enter after typing instead of pushing the submit button.

Please see formatting instructions below the problem.

Prove the identity sec(−x)tan(x)+cot(x)=−sin(−x)sec(-x)tan(x)+cot(x)=-sin(-x)

We will work on the left side:

sec(−x)tan(x)+cot(x)sec(-x)tan(x)+cot(x)

==

Use an even/odd identity:

==

tan(x)+cot(x)tan(x)+cot(x)

Now you have:

/(tan(x)+cot(x))/(tan(x)+cot(x))

==

Use reciprocal and quotient identities on all 3 terms:

==

++

Now you have:

()+()()+()

==

Get a common denominator and add the 2 fractions in the denominator:

==

Now you have:

==

Use a pythagorean identity:

==

sin(x)cos(x)sin(x)cos(x)

Now you have:

/(sin(x)cos(x))/(sin(x)cos(x))

==

Rewrite the division of fractions as the numerator fraction multiplied by the reciprocal of the denominator fraction:

==

Now you have:

==

Cancel:

==

Now you have:

==

Use even/odd identity:

=

Proven! Now you have:

FORMATTING INSTRUCTIONS BELOW

Type in lowercase and use parentheses. For example type sin(x) not sinx.

For fractions use the divide symbol "/". For example sin(x)/cos(x) for sin(x)cos(x)sin(x)cos(x)

For multiplication of fractions use parentheses. For example (cos(a)/sin(b))(tan(c)/sin(d)) for (cos(a)sin(b))(tan(c)sin(d))(cos(a)sin(b))(tan(c)sin(d)).

For multiplication without fractions no symbol is necessary. For example cos(a)sin(b)

For exponents use "^". Parentheses will be necessary depending on how you choose to type your answer For example sin^2(x) for sin2(x)sin2(x). Or (sin(x))^2 for (sin(x))2(sin(x))2.

For fractions with multiple terms in the numerator or denominator parentheses are necessary. For example (1+sin(x))/cos(x) for 1+sin(x)cos(x)1+sin(x)cos(x). Or (1+sin(x))/(1-sin(x)) for 1+sin(x)1−sin(x)1+sin(x)1-sin(x)