tanx cotx
------- + ------- = 1 +secxcscx
1 - cotx 1- tanx
prove the proof
tanx/(1-cotx) + cotx/(1-tanx)
= tanx/(1-cotx) + cotx/(1-tanx) * (-tanx/-tanx)
= tanx/(1-cotx) - tan^2x/(1-tanx)
= (cotx-tan^2x)/(1-tanx)
= (1-tan^3x)/(tanx(1-tanx))
= (1-tanx)(1+tanx+tan^2x) / (1-tanx)(tanx)
= (1+tanx+tan^2x)/tanx
= 1/tanx + 1 + tanx
= 1 + sinx/cosx + cosx/sinx
= 1 + (sin^2x+cos^2x)/(sinx cosx)
= 1 + secx cscx
oops . typo in line 2, but you should be able to catch it.
To prove this equation, we will start by simplifying the left side of the equation and the right side of the equation separately. Then, we will compare them to check if they are equal.
Let's simplify the left side of the equation:
Step 1: Simplify the denominator on the left side.
Denominator = (1 - cotx)
We can rewrite cotx as 1/tanx:
Denominator = (1 - 1/tanx)
To simplify the denominator, we multiply both terms by tanx:
Denominator = (tanx - 1)/tanx
Step 2: Simplify the numerator on the left side.
Numerator = tanx
The numerator does not require further simplification.
Combining the numerator and denominator, we have:
(tanx) / [(tanx - 1)/tanx]
To simplify this expression further, we can rationalize the denominator by multiplying both the numerator and denominator by tanx:
(tanx * tanx) / (tanx - 1)
Now, let's simplify the right side of the equation:
Step 1: Simplify the denominator on the right side.
Denominator = (1 - tanx)
The denominator does not require further simplification.
Step 2: Simplify the numerator on the right side.
Numerator = (1 + secx * cscx)
We can rewrite secx as 1/cosx and cscx as 1/sinx:
Numerator = (1 + (1/cosx) * (1/sinx))
= (1 + (sinx * cosx)/(cosx * sinx))
= (1 + 1)
= 2
Combining the numerator and denominator, we have:
(2) / (1 - tanx)
Now, let's compare the left side and the right side of the equation:
(tanx * tanx) / (tanx - 1) = 2 / (1 - tanx)
To show that they are equal (prove the equation), we will cross-multiply:
(tanx * tanx) * (1 - tanx) = 2 * (tanx - 1)
Expanding both sides:
tanx * tanx - tanx^3 = 2 * tanx - 2
Rearranging:
tanx * tanx - 2 * tanx + tanx^3 - 2 = 0
Factoring:
(tanx - 1) * (tanx^2 + tanx - 2) = 0
Simplifying:
(tanx - 1) * (tanx + 2) * (tanx - 1) = 0
From this equation, we can see that (tanx - 1) = 0 or (tanx + 2) = 0.
If (tanx - 1) = 0, then tanx = 1.
If (tanx + 2) = 0, then tanx = -2.
However, tanx = -2 is not possible since the range of the tangent function is (-∞, ∞). Therefore, it is only valid when tanx = 1.
Hence, we have proved that the given equation is true:
tanx / (1 - cotx) + cotx / (1 - tanx) = 1 + secx * cscx.