Help me with this........I cannot understand how to simplify?

Question

Evaluate:

sin(-t) * csc(-t) + cos(t) * sec(-t)

--

Evaluate:

Sin (5/8 pi)

Answer 1

cosec(x) = 1/sin(x)
& sec(x) = 1/cos(x)
& cos(-x) = cos(x)

then,

sin(-t)*cosec(-t) + cos(t) * sec(-t)
= sin(-t)*1/sin(x) + cos(t) * 1/cos(-t)
= 2

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Sin(5/8 pi)
to calculate its value, use:

cos(2x) = 1 - 2[sin(x)]^2
then,
cos(5/4 pi) = 1 - 2[Sin(5/8 pi)]^2
[Sin(5/8 pi)]^2 = 0.5 [ 1 - cos(5/4 pi) ]
but cos(5/4 pi) = - cos(1/4 pi) = -1/√2

then,
[Sin(5/8 pi)]^2 = 0.5 (1 + 1/√2)
or
Sin(5/8 pi) = √[0.5(1 + 1/√2)]

Answer 2

go to the definitions of the trig functions:
csc(-t) = 1/sin(-t) and sec(-t) = 1 /cos(-t)

now substitute the into given expression:
sin(-t)* 1/sin(-t) + cos(t)* 1/cos(-t)
simplify:
sin(-t)/sin(-t) + cos(t)/cos(-t)
but cos(t) = cos(-t)
sin(-t)/sin(-t) + cos(t)/cos(t)
1 + 1 = 2


sin (5/8pi) pi radians = 180 degrees therefore
5/8 pi = 5*180/8 = 112.5 degrees
sin(112.5) = ? look this up in a math table or use your calculator

Answer 3

csc = 1/sin and sec=1/cos.

Moreover, cos(t)=cos(-t) so you should be able to simplify all this to a nice small number.

Answer 4

Question 1
sin(-t) x cosec (-t) + cos t x sec(-t)

= -sin t x 1 / sin(-t) + cos t x 1/ (cos(-t)

= -sin t x 1/ (- sint) + cos t x 1/ cos t

= 1 + 1

= 2

Question 2
sin[ (5/8 . π] = sin(112.5°) = 0.924

It may be however that you mean:-

sin[ 5/(8π] = sin (11.4°) = 0.198

Answer 5

csc = 1/sin
sec = 1/cos
cos(t)=cos(-t)

sin(-t) * 1/(sin(-t) + cos(t) * 1/cos(-t)
1 + cos(t)/cos(-t)
1+1
=2