True or False. Proving Identities

Question

(1 +sin t)/cos t + (cos t)/(1 + sin t) = (2 csc t) is an identity.

True or False?

Answer 1

False

Lets work on the left side

combine the 2 fractions ( common denominator is (cos)(1+sin) and you get

(1+2sin +sin^2 +cos^2)/ (1+sin)(cos). Since sin^2+cos^2 =1this simplifies to

2+2sin/(1+sin )cos)

factor 2 out of the numerator to give you 2(1+sin)/(1+sin)(cos) which then simplifies to

2/cos

2/cos is not equal to 2 csc as 2csc is 2/sin

2/cos does not equal 2/sin

Answer 2

Here is the equation: (1 +sin t)/cos t + (cos t)/(1 + sin t) = (2 csc t)

First step is to find common denominator of left side. That will be (cos t)(1+ sin t)

[(1+sin t)(1+sin t)]/ [(cos t)(1+sin t)] + [(cos t)(cos t)] /[(cos t)(1+sin t)]

Use FOIL to expand numerator of first fraction and add to numerator of 2nd fraction: [1+2 sin t +sin^2 (t) + cos^2 (t)] / [(cos t) (1 + sin t)]

Substituting for sin^2 (t) + cos^2 (t) = 1

[1+2 sin t +1] /[ (cos t)(1+sin t)]

[ 2+ 2sin t] / [(cos t)(1+sin t)]

Factor out a 2 yields the following:

2(1+sin t)/[(cos t)(1+ sin t)]

Cancelling out (1 + sin t) from top and bottom yields:

2/ (cos t) is equivalent to 2 sec (t). Therefore the problem statement is False.