Question

1 + sec(-x) over

sin(-x) + tan(-x)
=-csc x
Please help!! This a fraction I don't know how to do

Answer 1

Breathing rate estimation

Liban Islam

On average, an adult at rest breathes in and out 12 times per

minute. About how many breaths will an adult at rest take in a

day?

There are different ways to approach this problem, but one possible method is:

One minute has 60 seconds, so an adult at rest breathes in and out 12/60 = 0.2 times per second.

One hour has 60 minutes, so an adult at rest breathes in and out 0.2 x 60 = 12 times per hour.

One day has 24 hours, so an adult at rest breathes in and out 12 x 24 = 288 times per day.

Therefore, an adult at rest would take about 288 breaths per day on average. It's worth noting that this is just an approximation and actual breathing rates may vary depending on factors such as age, health, activity level, and environment.

Liban Islam

1 + sec(-x) over

sin(-x) + tan(-x)

=-csc x

We can start by using the identities:

sec(-x) = sec(x), since the secant function is even.

sin(-x) = -sin(x), since the sine function is odd.

tan(-x) = -tan(x), since the tangent function is odd.

csc(x) = 1/sin(x), since the cosecant function is the reciprocal of the sine function.

Using these identities, we can rewrite the left-hand side of the equation as:

1 + sec(-x) / sin(-x) + tan(-x)

= 1 + sec(x) / (-sin(x)) + (-tan(x)) (substituting the identities)

= 1 - (1/cos(x)) / (sin(x)/cos(x)) - tan(x) (rearranging and simplifying)

= 1 - (1/cos(x)) * (1/sin(x)) - tan(x) (multiplying by reciprocal)

= 1 - csc(x) - tan(x) (using the identity csc(x) = 1/sin(x))

Now, we can use the identity tan(x) = sin(x) / cos(x) and rearrange to get:

1 + sec(-x) / sin(-x) + tan(-x)

= 1 - csc(x) - tan(x)

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

= (cos(x) - sin(x)^2) / (sin(x) cos(x))

= cos(x)/sin(x) - sin(x)/sin(x) * sin(x)/cos(x)

= cot(x) - tan(x)

= -csc(x)

Therefore, we have shown that:

1 + sec(-x) / sin(-x) + tan(-x) = -csc(x)

for all x such that sin(x) is not equal to zero (i.e., x is not an odd multiple of pi/2).

Answer 2

To solve this problem, we will use the following trigonometric identities:

sin(-x) = -sin(x)

tan(-x) = -tan(x)

sec(x) = 1/cos(x)

csc(x) = 1/sin(x)

Explanation:

First, let's simplify the numerator using the identity sec(-x) = 1/cos(-x) = 1/cos(x):

1 + sec(-x) = 1 + 1/cos(x)

Next, let's simplify the denominator using the identities sin(-x) = -sin(x) and tan(-x) = -tan(x):

sin(-x) + tan(-x) = -sin(x)/cos(x) - tan(x)/cos(x)

= (-sin(x) - tan(x))/cos(x)

Now we can substitute these simplified expressions into our original fraction:

(1 + sec(-x)) / (sin(-x) + tan(-x))

= [1 + 1/cos(x)] / [(-sin(x) - tan(x))/cos(x)]

To simplify further, we can flip the denominator and multiply by the reciprocal:

[1 + 1/cos(x)] * [-cos(x)/(sin(x) + tan(x))]

= [-cos(x) + cos(x)/cos(x)] / [sin(x)/cos(x) - sin(x)/cos(x) + (-sin(x)/cos(x))*tan(x)]

= [-cos(x) + 1] / [-sin(x)/cos(x) * (1 - tan(x))]

= [-cos(x) + 1] / [-sin(x)/cos(x) * (-sin(x)/cos(x))]

= [-cos(x) + 1] / [sin^2(x)/cos^2(x)]

= [-cos(x) + 1] * cos^2(x)/sin^2(x)

= (-cos^3(x) + cos^2(x))/sin^2(x)

= cos^2(x)(-cos(x) + 1)/sin^2(x)

Finally, we can use the identity csc(x) = 1/sin(x) to simplify:

cos^2(x)(-cos(x) + 1)/sin^2(x) = -cos^2(x)/sin^2(x) + cos^2(x)/sin^2(x) = -1/sin^2(x) = -csc^2(x)

Therefore,

(1 + sec(-x)) / (sin(-x) + tan(-x)) = -csc^2(x)

or, taking the square root of both sides,

(1 + sec(-x)) / (sin(-x) + tan(-x)) = -csc(x)