For this question, they want me to use fundamental trig identities to simplify the expression. The problem is as follows; (tanx/csc^2x + tanx/sec^2x)(1+tanx/1+cotx) - 1/cos^2x

Question

For this question, they want me to use fundamental trig identities to simplify the expression. The problem is as follows; (tanx/csc^2x + tanx/sec^2x)(1+tanx/1+cotx) - 1/cos^2x

I got as far as this; tanx(1/csc^2x + 1/sec^2x)(1+tanx/1+cotx) - sec^2x. I factored out the tangent and simplified the 1/cos^2x to sec^2x. Then I simplified further by saying that tanx(sin^2x+cos^2x)((1+tanx/1+cotx)-sec^2x. just not sure how to simplify down the 1+tanx/1+cotx. Some help would be much obliged

Answer 1

(1+tanx)/(1+cotx) = tanx(1+tanx) / tanx(1+cotx)

= tanx(1+tanx) / (1+tanx)
= tanx

Answer 2

I am going to insert some necessary brackets where I think they probably should be:

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

check your typing, this does not reduce to the answer you stated.

Answer 3

Ok, picking up from oobleck's

(1+tanx)/(1+cotx) = tanx, we get

tanx (tanx) - tan^2 x -1
= -1 , which would not be the answer you gave.

Answer 4

i never gave a final answer just as far as i got.

oobleck are you able to explain what identities you used to break down that part? I'm just not seeing how it was simpified

Answer 5

looks like oobleck is not online, so I will explain

As he has shown, he has multiplied top and bottom by tan x
(1+tanx)/(1+cotx) = tanx(1+tanx) / tanx(1+cotx)
left the top as is, but expanded the bottom, realize that tanxcotx = 1
= tanx(1+tanx) / (1+tanx)

so the bottom becomes tanx + 1, cancels the top 1+tanx, leaving tanx

Answer 6

To simplify the expression further, we'll start by simplifying the expression 1+tanx/1+cotx.

We can rewrite tanx as sinx/cosx and cotx as cosx/sinx. Substituting these expressions, we have:

1 + (sinx/cosx) / [1 + (cosx/sinx)]

To simplify this, we'll get a common denominator by multiplying both the numerator and denominator of the fraction inside the brackets by sinx:

1 + (sinx/cosx) * sinx / [1 * sinx + cosx * sinx]

This gives us:

1 + sin^2x / (sinx + cos^2x)

Using the Pythagorean identity sin^2x + cos^2x = 1, we can rewrite the denominator as 1:

1 + sin^2x / 1

Simplifying further, we have:

1 + sin^2x

Now, returning to the original expression, we have:

tanx * [(sin^2x + cos^2x) * (1 + sin^2x) - sec^2x]

Since sin^2x + cos^2x = 1, the expression becomes:

tanx * (1 * (1 + sin^2x) - sec^2x)

Expanding, we have:

tanx * (1 + sin^2x - sec^2x)

Using the fundamental trigonometric identity sec^2x = 1 + tan^2x, we can rewrite the expression as:

tanx * (1 + sin^2x - (1 + tan^2x))

Simplifying, we have:

tanx * (1 + sin^2x - 1 - tan^2x)

Combining like terms, we get:

tanx * (sin^2x - tan^2x)