Prove cotx-1/cotx+1 = sec2x - tan2x
I prove till cotx-1/cotx+1 =1/1+tanx - tanx/1+tanx
To prove that cotx-1/cotx+1 = sec2x - tan2x, we need to start with the left side of the equation and simplify it step by step until we reach the right side of the equation.
1. Start with the left side of the equation: cotx-1/cotx+1.
2. Rewrite cotx-1 as (1/tanx)-1.
3. Rewrite cotx+1 as (1/tanx)+1.
So now we have (1/tanx)-1 / (1/tanx)+1.
4. Multiply the numerator and the denominator by (tanx+1) to get rid of the fractions.
[(1/tanx)-1]*(tanx+1) / [(1/tanx)+1]*(tanx+1).
This gives us (tanx+1)/(tanx-1) / (1+tanx)/(1/tanx+1).
5. Rewrite (tanx+1)/(tanx-1) as [(tanx+1)/(tanx-1)]*[(1/tanx+1)/(1/tanx+1)].
Now we have [(tanx+1)*(1/tanx+1)] / [(tanx-1)*(1/tanx+1)].
6. Simplify the numerator:
(tanx+1)*(1/tanx+1) = 1.
7. Simplify the denominator:
(tanx-1)*(1/tanx+1) = (tanx-1)/(tanx+1).
8. So now we have 1 / (tanx-1)/(tanx+1).
9. Rewrite (tanx-1)/(tanx+1) as [(tanx+1-tanx)/(tanx+1)].
So now we have 1 / (1 - tanx)/(tanx+1).
10. Multiply the numerator and denominator by (tanx+1) to get rid of the fraction.
1*(tanx+1) / [(1 - tanx)/(tanx+1)] * (tanx+1).
This gives us (tanx+1) / (1 - tanx).
11. Now we need to simplify the right side of the equation, which is sec2x - tan2x.
Sec2x is equal to 1/cos2x, and tan2x is equal to sin2x/cos2x.
So sec2x - tan2x can be simplified as: 1/cos2x - sin2x/cos2x.
12. Combine the fractions with the same denominator:
(1 - sin2x)/cos2x.
13. Use the trigonometric identity sin2x = 1 - cos2x to simplify further:
(1 - (1 - cos2x))/cos2x.
14. Simplify the numerator: (1 - 1 + cos2x)/cos2x.
This gives us cos2x/cos2x, which simplifies to 1.
Therefore, the left side of the equation (cotx-1/cotx+1) simplifies to 1, which is also the right side of the equation (sec2x - tan2x). This proves that cotx-1/cotx+1 = sec2x - tan2x.
To prove: cotx-1/cotx+1 = sec^2x - tan^2x
We'll start with the left side of the equation and simplify it step by step to see if it matches the right side of the equation.
1. Start with the left side of the equation:
cotx-1/cotx+1
2. Multiply the numerator and denominator by cotx+1:
(cotx-1)(cotx+1) / (cotx+1)(cotx+1)
3. Expand both the numerator and denominator:
(cot^2x - 1) / (cot^2x + 2cotx + 1)
4. Use the cotangent identity: cot^2x + 1 = csc^2x
(csc^2x - 1) / (csc^2x + 2cotx + 1)
5. Rewrite cotx as 1/sinx:
(csc^2x - 1) / (csc^2x + 2/sinx + 1)
6. Rewrite csc^2x as 1/sin^2x:
(1/sin^2x - 1) / (1/sin^2x + 2/sinx + 1)
7. Combine the fractions in the numerator:
(1 - sin^2x) / (sin^2x + 2sinx + 1)
8. Use the trigonometric identity: 1 - sin^2x = cos^2x
cos^2x / (sin^2x + 2sinx + 1)
9. Rewrite sin^2x as 1 - cos^2x:
cos^2x / (1 - cos^2x + 2sinx + 1)
10. Rearrange the terms:
cos^2x / (2sinx - cos^2x + 2)
11. Use the Pythagorean identity: 1 - sin^2x = cos^2x
cos^2x / (2sinx + sin^2x + 1)
12. Simplify the expression: sin^2x + cos^2x = 1
cos^2x / (sin^2x + 2sinx + 1)
13. Use the reciprocal identity: sin^2x = 1 - cos^2x
cos^2x / ((1 - cos^2x) + 2sinx + 1)
14. Simplify the expression in the denominator: (1 - cos^2x) + 2sinx + 1 = 2 + 2sinx
cos^2x / (2 + 2sinx)
15. Use the double angle identity: sin(2x) = 2sinx*cosx
cos^2x / (2 + 2sinx*cosx)
16. Use the Pythagorean identity: cos^2x = 1 - sin^2x
(1 - sin^2x) / (2 + 2sinx*cosx)
17. Use the reciprocal identity: cosx = 1/sinx
(1 - sin^2x) / (2 + 2sinx/sinx)
18. Combine the fractions in the denominator: 2sinx/sinx = 2
(1 - sin^2x) / (2 + 2)
19. Simplify the expression: 2 + 2 = 4
(1 - sin^2x) / 4
20. Use the Pythagorean identity: sin^2x + cos^2x = 1
(1 - sin^2x) / 4 = cos^2x / 4
21. Simplify the expression: cos^2x / 4 = (1/cos^2x) * (cos^2x / 4) = (1/4)sec^2x
Therefore, we've shown that cotx-1/cotx+1 = sec^2x - tan^2x.
Note: In this proof, we've used various trigonometric identities to simplify the expressions step by step. By manipulating the expressions using known identities, the left side was transformed into the right side of the equation.
For this to work you must mean
(cotx-1)/(cotx+1) = sec(2x) - tan(2x)
LS = (cosx/sinx - 1)/(cosx/sinx +1)
multiply top and bottom by sinx
= (cosx - sinx)/(cosx + sinx)
RS = 1/cos(2x) - sin(2x)/cos(2x)
= (1 - 2sinxcosx)/(cos^2 x - sin^2 x)
= (sin^2 x + cos^2 x - 2sinxcosx)/((cosx+sinx)(cosx-sinx))
= (sinx - cosx)^2/( (cosx+sinx)(cosx-sinx))
= (sinx - cosx)/(cosx + sinx)
= LS