Question

Please help me verify this identity: (1+tanx)/(1+cotx) = (secx)/(cscx)

Answer 1

Hi

Recall that:
cot(x) = 1/tan(x), 1/sin(x) = csc(x), 1/cos(x) = sec(x)………(1) (reciprocal identities)
tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)………(2) (quotient identities)
cos^2(x) + sin^2(x) = 1………(3) (pythagorean identity)

[1 - tan(x)]*[1 - cot(x)]
= 1*1 + 1*-cot(x) - tan(x)*1 - tan(x)*-cot(x) <--- by expanding it out
= 1 - cot(x) - tan(x) + tan(x)cot(x)
= 1 - cot(x) - tan(x) + tan(x)/tan(x) <--- by (1)
= 1 - cot(x) - tan(x) + 1
= 2 - cot(x) - tan(x)
= 2 - cos(x)/sin(x) - sin(x)/cos(x) <--- by (2)
= 2 - cos^2(x)/[sin(x)cos(x)] - sin^2(x)/[sin(x)cos(x)] <--- common denominator
= 2 - [cos^2(x) + sin^2(x)]/[sin(x)cos(x)]
= 2 - 1/[sin(x)cos(x)] <--- by (3)
= 2 - [1/sin(x)]*[1/cos(x)]
= 2 - csc(x)*sec(x) <--- by (2)
= 2 - sec(x)csc(x)

Answer 2

@callmemaybe, i'm alittle confused at how you did the denominator, and i think you changed the + in the numerator to a -. Thank you for your help

Answer 3

I would go along this lines:

\(\dfrac{1+\dfrac{\sin x}{\cos x}}{1+\dfrac{\cos x}{\sin x}}=\dfrac{\dfrac{\cos x}{\cos x}+\dfrac{\sin x }{\cos x}}{\dfrac{\sin x}{\sin x}+\dfrac{\cos x}{\sin x}}=\dfrac{\dfrac{\cos x+\sin x}{\cos x}}{\dfrac{\sin x+\cos x}{\sin x}}=\dfrac{\cos x+\sin x}{\cos x}\cdot \dfrac{\sin x}{\cos x+\sin x}=\)


\(=\dfrac{\sin x}{\cos x}\)

Answer 4

Now, this could be just \(\tan x\), but we have to show something else:

it is supposed to be \(\dfrac{\sec x}{\csc x}\)...

Let's try: \(\dfrac{\sin x}{\cos x}=\dfrac{1}{\cos x}\cdot \sin x=\dfrac{1}{\cos x}: \dfrac{1}{\sin x}=\sec x : \csc x = \dfrac{\sec x}{\csc x}\)

Answer 5

in your first response, how did you go from step 3 to 4?

Answer 6

I used: \(\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=\dfrac{a}{b} \cdot \dfrac{d}{c}\)

Answer 7

ok, i did that and got sin/cos. Now i'll try to understand your part 2

Answer 8

what does the colon mean?

Answer 9

It is another notation for division :)

Answer 10

i never would have come up with this on my own, thank you so much