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Complete the identity: csc(t)(sin(t) + cos(t)) = ?

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4 Answers

Whenever you have a complicated problem like this, it is best to break it down into Sines and Cosines- so 

csc(t)(sin(t)+cos(t))  since csc(t) = 1/sin(t), we have 

(sin(t)+cos(t))*1/(sin(t) Distributing out, we get

sin(t)/sin(t) + cos(t)/sin(t)    This simplifies to

1 + cot(t)

Recall #1     cst t =1/sin t , cst t  is the inverse of sin t

          #2     cos t = cos(t/2 + t/2)=cos t/2 cos t/2 - sin t/2 sin t/2 using the formula for cos(a+b)

                    where a=b=t/2

          #3      1 = cos t/2 cos t/2 + sin t/2 sin t/2  , The fundamental relationship in Trigonometry

                     rewritten with minor acceptable changes.

Therefore, csc t sin t + cos t = [1/(sin t/2)] sin t/2 + [cos t/2 cos t/2  - sin t/2 sin t/2]

                                          = 1 + cos t/2 cos t/2 - sin t/2 sin t/2

                                          = cos t/2 cos t/2 + sin t/2 sin t/2 + cos t/2 cos t/2 - sin t/2 sin t/2

                                          = 2 [ cos t/2 cos t/2 ]

                                          = 2 [ cos t/2 ]^2 , read 2 multiplied by (cos t/2) squarred!