Prove the following identity: 1 − cos ⁡ x sin ⁡ x + sin ⁡ x 1 − cos ⁡ x = 2 csc ⁡ x

Question

Prove the following identity:             1      −      cos      ⁡      x              sin      ⁡      x        +            sin      ⁡      x              1      −      cos      ⁡      x        =  2  csc  ⁡  x

Heidy Phelps · Accepted Answer

L.H.S.  [  (  1  −  cos  ⁡  x  )      /    sin  ⁡  x  ]  +  [  sin  ⁡  x      /    (  1  −  cos  ⁡  x  )  ]  =  [  (  1  −  cos  ⁡  x  )  ∗  (  1  −  cos  ⁡  x  )  +  (  sin  ⁡  x  ∗  sin  ⁡  x  )  ]      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ]....doingL.C.M.  =  [  1  +      cos          2        ⁡  x  −  2  cos  ⁡  x  +      sin          2        ⁡  x  ]      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ]  =  [  1  +      cos          2        ⁡  x  +      sin          2        ⁡  x  −  2  cos  ⁡  x  ]      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ]  =  [  1  +  1  −  2  cos  ⁡  x  ]      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ] .......as       sin          2        ⁡  x  +      cos          2        ⁡  x  =  1  =  [  2  −  2  cos  ⁡  x  ]      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ]  =  2  ∗  (  1  −  cos  ⁡  x  )      /    [  sin  ⁡  x  ∗  (  1  −  cos  ⁡  x  )  ]  =  2      /    sin  ⁡  x=2cosec x ......as   1      /    sin  ⁡  x  =  c  o  s  e  c  x= R.H.S.hence L.H.S. = R.H.S.= proved

Prove the following identity: (1-cos x)/(sin x)+ (sin x)/(1-cos x)=2csc x

Answered question

Answer & Explanation

New Questions in Trigonometry