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

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

### 4 Answers by Expert Tutors

Xavier J. | Tutor in Math, topics range from Algebra to Calculus.Tutor in Math, topics range from Algebra...
5.0 5.0 (3 lesson ratings) (3)
1

Keeping in that csc(t) = 1/sin(t), distribute that csc(t) and you get csc(t)(sin(t) + cos(t)) = 1 + cot(t)

Yitzhak S. | Johns Hopkins Engineer, Specializing in Math and Science.Johns Hopkins Engineer, Specializing in ...
4.8 4.8 (4 lesson ratings) (4)
0

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)

George C. | Humboldt State and Georgetown graduateHumboldt State and Georgetown graduate
5.0 5.0 (2 lesson ratings) (2)
0

You have a typo here.

Bam K. | Multi-discipline, Passionate, and Compassionate TutorMulti-discipline, Passionate, and Compas...
4.5 4.5 (12 lesson ratings) (12)
-1

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!