Question

tantheeta/1-cottheta+cottheeta/1-tantheeta prove that =1+tantheeta+cottheeta

Answer 1

As the Question is,

tanФ/1-cotФ + cotФ/1-tanФ = 1 + tanФ + cotФ


Taking LHS

= tanФ/1-cotФ + cotФ/1-tanФ

= tanФ/1-cotФ + cotФ ÷ 1-1/cotФ             

(tanФ = 1/cotФ)

= tanФ/1-cotФ + cot²Ф/cotФ-1   
           
(Taking LCM and transposing cotФ to numerator)

= tanФ/1-cotФ - cot²Ф/1-cotФ   

(we make - cot²Ф to bring same denominator)

= 1/cotФ - cot²Ф ÷ 1 - cotФ

(tanФ = 1/cotФ)

= 1 - cot³Ф/cotФ(1-cotФ)

(Taking LCM and bringing cotФ to denominator)

= (1-cotФ)(1+cot²Ф+cotФ)/cotФ(1-cotФ)

(From formula a³ - b³ = (a-b)(a²+b²+ab) )

= 1/cotФ + cot²/cotФ + cotФ/cotФ

(Cancelling (1-cotФ) on both numerator and denominator, Also take cotФ separately in order to simplify easily)

= tanФ + cotФ + 1

= 1 + tanФ + cotФ  = RHS


HENCE PROVED

☺Hope this Helps☺

Answer 2

s ur question is proved