Simplify The Equation As Much As Possible

Question

Simplify the following expressions as much as possible:

1.(1/1-Sin(T))+(1/1+Sin(T)) = ?

2.(1/Sin(T)*Cos(T))-(1/Tan(T)) = ?

Simplify completely into an expression with Sin(A) or Cos(A) only:

3.Sin(A)*Tan(A)+Cos(A) = ?

Answer 1

Hi Miguelina!

I'll go through the first problem and will give you

some tips to solve the other two.

1 1

------- + --------

1-SinT 1+SinT

Find the LCD. It is (1-SinT)(1+SinT).

(1+SinT) 1 (1-SinT) 1

------------ * ------- + ------------ * ---------

(1+SinT) 1-SinT (1-SinT) 1+SinT

(1+SinT) (1-SinT)

---------------------- + ---------------------

(1+SinT)(1-SinT) (1+SinT)(1-SinT)

(SinT - SinT cancels out in the numerator)

2

-------------------------------- Middle terms cancel out

1-SinT+SinT-Sin²T in the denominator

2 Pythagorean Identity

--------------------------- Sin²T + Cos²T = 1

1 - Sin²T or Cos²T = 1 - Sin²T

2

--------------

Cos²T

1

2Sec²T (Sec²T = ------- )

Cos²T

For the second problem, note that

CosT 1

------- = CotT and that CotT = --------

SinT TanT

For the third problem, note that

SinA

TanA = --------

CosA

Get a common denominator of CosA, then

you can use the pythagorean identity again

to simplify the numerator.

Hope you found that helpful!

Answer 2

1) Get a common denominator:

2) Convert the tan(θ) into sin(θ)/cos(θ) then again get a common denominator.

3) Again, convert Tan(A) into sin(A)/cos(A) then again, get a common denominator.

Answer 3

All three make use of identity sin² + cos² = 1.

problem 1) Apply LCD = (1 - sin)*(1 + sin) [to save time, I'm leaving off the argument (T) until the last step]:

1/(1 - sin) * LCD + 1/(1 + sin) * LCD/LCD = (1 + sin) / (1 - sin²) + (1 - sin) / (1 - sin²) = 2/ cos² (T) [simplest = 2*sec² (T), but are problems 1 & 2 supposed to be left in terms of sin and cos only?].

problem 2) Apply LCD = sin*cos: 1/(sin*cos) - 1/tan = 1/(sin*cos) - cos/sin, times LCD/LCD --> 1/(sin*cos) - cos² /sin*cos = (1 - cos²) / sin*cos = sin² / sin*cos = sin(T) / cos(T) [ simplest = tan(T)].

problem 3) sin*tan + cos = sin*(sin/cos) + cos² /cos = (sin² + cos²) / cos = 1/cos(T).