Establish the identity.−tan t + cos t / 1−sin t = sec t.

Question

Write the left side in terms of sine and cosine. - ? + cos t / 1 - sin t.

Add fractions from the previous step. Apply the appropriate Pythagorean identity to simplify the numerator.

? / cos t(1 - sin t). (Simplify your answer.)

Divide any common factors out of the numerator and denominator from the previous step. Do not apply any trigonometric identity.

?

The fraction from the previous step then simplifies to sec t using​ what?

A. Even-Odd Identity.

B. Cancellation Property.

C. Quotient Identity.

D. Reciprocal Identity.

Answer 1

(-T+C)/(1-S) = 1/C

T=Tant S=Sint C=Cost, Sect = 1/Cost

and Tant = Sint/Cost, T=S/C

(-S/C +C)/(1-S) = 1/C

multiply both sides by C(1-S) to eliminate the fractions and get

-S +S^2 +C^2 = 1-S

replace C^2= 1-S^2

and cancel the -S terms on both sides

S^2 + 1-S^2 = 1

1=1

QED.

it's an identity, true for any angle

or you may have intended to write

-tant + (cost)/(1+sint) = sect? which is technically the way it's written

then

-sint/cost + cost/(1-sint) = 1/cost

multiply by cost

-sint + cos^2(t)/(1-sint) = 1

multiply by 1+sint

-sint -sin^2(t) + cos^2(t) = 1 + sint

replace cos^2(t) with 1-sin^2(t) (the Pythagorean Theorem cos^2(t) + sin^2(t) = 1)

-sint -sin^2(t) + 1 - sin^2(t) = 1 + sint

the 1's cancel leaving

-sint - 2sin^2(t) = sint

-2sin^2(t) -2sint = 0

divide by -1

2sin^2(t) +2sint = 0

divide by 2

sin^2(t) +sint = 0

sint(sint +1) = 0

sint = 0 or -1

t = 0, 180 or 270 degrees, or pi or 3pi/2 radians

which is not a trig identity

just one counterexample will prove it's not a trig identity

try t=45 degrees

(-tan45+cos45)/(1-sin45) = sec45

= about (-1 +.71)/(1-.71) = 1/.71

= (-.29)/.29 doesn't = 1/.71 because the sign is different

if an equation isn't true for all, it's Not an identity

try the other possible interpretation of the problem, with 45 degrees

-tant + cost/(1-sint) = sect

=-tan45 + cos45/(1-sin45) = sec45

-1 + .71/(1-.71) = 1/.71

-1 + .71/.29 = 1/.71

-1 + 2.4 = 1.4

it seems to work

try t= 30

-tan30 + cos30/(1-sin30) = sec30

-1/sqr3 + (sqr3/2)/(1-1/2) = 1/(sqr3/2)

-1/sqr3 + sqr3 = 2sqr3/3

-sqr3/3 + 3sqr3/3 = 2sqr3/3

2sqr3/3 = 2sqr3

It looks like a trig identity. so there's an error in the calculations above, somewhere. One way or the other.

Answer 2

-tant + cost/(1 - sint) = - sint/cost + cost/(1 - sint) = [- sint(1 - sint) + cos²t]/[cost(1 - sint)] = (- sint + sin²t + cos²t)/([cost(1 - sint)] = (1 - sint)/[cost(1 - sint)] = 1/cost = sect