solve each identity algebraically

1)(1-tanx)/(1-cotx)=-tanx

2)(1+cotx)/(1+tanx)=cotx

my usual approach is to start with the "messy" side, and change everything to sines and cosines.

LS
=(1- sinx/cosx)/(1 - cosx/sinx)
= (cosx-sinx)/cosx * (sinx)/ (sinx - cosx)
= (-1)sinx/cosx
= -tanx

perform similar steps for the second one, bet you'll get it

I got (1- sinx/cosx)/(1 - cosx/sinx)

but how did (1- sinx/cosx) turn to (cosx-sinx)/cosx?

nevermind, found out :) thanks

Sure! Let's solve each identity algebraically.

1) To solve the first identity, we'll start by simplifying the expression on the left-hand side (LHS) using trigonometric identities.

Remember that:

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

Now, let's simplify the LHS step by step:

(1 - tan(x))/(1 - cot(x))

We can rewrite tan(x) as sin(x)/cos(x), and cot(x) as cos(x)/sin(x):

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

Next, we'll find a common denominator for both terms:

[(cos(x) - sin(x))/cos(x)] / [(sin(x) - cos(x))/sin(x)]

To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:

[(cos(x) - sin(x))/cos(x)] * [sin(x)/(sin(x) - cos(x))]

Now, let's simplify further:

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

Using the identity sin^2(x) = 1 - cos^2(x), we can substitute it:

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

Expanding further:

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

Rearranging the terms:

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

Next, we'll use the identity cos^2(x) = 1 - sin^2(x):

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

Canceling out the "1" terms:

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

Finally, we'll use the identity sin(x)/cos(x) = tan(x) to simplify further:

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

Now we have:

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

And we can see that this expression equals -tan(x).

Hence, the first identity is algebraically solved.

2) Now let's solve the second identity:

(1 + cot(x))/(1 + tan(x))

Again, we'll use the trigonometric identities:

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

Now, let's simplify the expression:

(1 + cos(x)/sin(x))/(1 + sin(x)/cos(x))

Finding a common denominator:

[(sin(x) + cos(x))/sin(x)] / [(cos(x) + sin(x))/cos(x)]

Dividing fractions by multiplying the first fraction by the reciprocal of the second fraction:

[(sin(x) + cos(x))/sin(x)] * [cos(x)/(cos(x) + sin(x))]

Simplifying further:

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

Using the identity cos^2(x) = 1 - sin^2(x) to substitute:

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

Expanding:

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

Rearranging:

(cos(x)sin(x) + 1)/(sin(x)(cos(x) + sin(x)))

Using the identity sin(x)/cos(x) = tan(x):

(tan(x)cos(x) + 1)/(sin(x)(cos(x) + sin(x)))

Finally, we can see that this expression equals cot(x).

Thus, the second identity is algebraically solved.