simple trignometry question?

Question

tan(a)/1-cot(a)+cot(a)/1-tan(a)
prove = tan(a)+cot(a)+1

Answer 1

multiply first fraction by sin/sin and second by cos/cos [on left], and turn tan into sin/cos and cot into cos/sin. get

(sin²/cos)/(sin - cos) + (cos²/sin)/(cos - sin)

reverse sign of 2nd fraction and order of 2nd denominator, get

(sin²/cos)/(sin - cos) - (cos²/sin)/(sin - cos)

add tops, get

[ sin²/cos - cos²/sin] / (sin - cos)

add top and get

[(sin^3 - cos^3) / (sin cos)] / (sin - cos)

factor top

[(sin - cos)(sin² + sin cos + cos²)/(sin cos)] / (sin - cos)

cancel common (sin - cos)

(sin² + sin cos + cos²) / (sin cos)

and now divide through

sin/cos + 1 + cos/sin = tan + 1 + cot

Answer 2

Prove the identity.

tan(a)/(1-cot(a)) + cot(a)/(1-tan(a)) = tan(a)+cot(a)+1

Let's work with the left hand side.

tan(a)/(1-cot(a)) + cot(a)/(1-tan(a))
= (sin a/cos a)/(1 - cos a/sin a) + (cos a/sin a)/(1 - sin a/cos a)
= (sin² a/cos a)/(sin a - cos a) + (cos² a/sin a)/(cos a - sin a)
= (sin² a/cos a)/(sin a - cos a) - (cos² a/sin a)/(sin a - cos a)
= {(sin² a/cos a) - (cos² a/sin a)}/(sin a - cos a)
= (sin³ a - cos³ a)/{(sin a)(cos a)(sin a - cos a)}
= (sin a - cos a)(sin² a + (sin a)(cos a) + cos² a)
/{(sin a)(cos a)(sin a - cos a)}
= (sin² a + (sin a)(cos a) + cos² a)/{(sin a)(cos a)}
= (sin a/cos a) + 1 + (cos a/sin a)
= tan a + 1 + cot a
= tan(a) + cot(a) + 1 = right hand side

Answer 3

write first term as tan^2(a)/tan(a)-1 by putting cot(a)=1/tan(a)
we get the final expression as cot(a)-tan^2(a)/1-tan(a)
which can be written as 1-tan^3(a)/tan(a){1-tan(a)}
1-tan^3(a) is expanded as {1-tan(a)}{1+tan(a)+tan^2(a)}
divide this by denominator i.e. tan(a){1-tan(a)}
we get answer as
tan(a)+cot(a)+1

Answer 4

this should do it. for the top equation, add the two fractions together using a common denominator, then split up the fraction, then simplify. remember, when working with trig identities, you cannot cross the equal sign. in other words, you can't set them equal to each other, and then solve. you must solve to prove they are equal to each other.

Answer 5

Part of your question got cut off.

Answer 6

ohh