how do you veriy
tan x + Cot y
-------------- = tan y + cot x
1-tan x cot y
sin^4x + cos^4x = 1- 2cos^2x + 2cos^4
I get on the right:
sin y / cos y + cos x / sin x
(sin x sin y + cos x cos y) / sin x cos y
sin x sin y/cos x + 1 cos y
----------------------------
sin x cos y/cos x
tan x sin y + cos y
---------------------
tan x cos y
tan x sin y/sin y + cos y/sin y
--------------------------------
tan x cos y/sin y
tan x + cot y
---------------
tan x cot y
sin^4x + cos^4x = 1- 2cos^2x + 2cos^4 this is not complete, assume you mean
2 cos^4 x at the end
sin^4 x = sin^2 x * sin^2 x
=(1 - cos^2 x)(1 - cos^2 x)
= well I guess that gives it away
To verify the given equations, you need to simplify both sides of the equations separately and confirm whether they are equal. Let's start with the first equation:
tan x + Cot y
-------------- = tan y + cot x
1-tan x cot y
1. Simplify the left-hand side of the equation:
The left-hand side can be written as follows:
(tan x + Cot y) / (1 - tan x cot y)
Using the reciprocal identities for tangent and cotangent, the expression becomes:
(sin x / cos x + cos y / sin y) / (1 - (sin x / cos x) * (cos y / sin y))
2. Further simplification:
To simplify the expression, we need a common denominator. Multiply the numerator and denominator by sin x * sin y to obtain:
(sin x * sin y / cos x * sin y + cos y * cos x / sin y * cos x) / (sin x * sin y / cos x * sin y - cos y * cos x / sin y * cos x)
3. Simplify the expression in the numerator:
Use the distributive property to expand the numerator:
sin x * sin y / cos x * sin y + cos y * cos x / sin y * cos x =
(sin x * sin y + cos x * cos y) / (cos x * sin y)
4. Simplify the expression in the denominator:
sin x * sin y / cos x * sin y - cos y * cos x / sin y * cos x =
(sin x * sin y - cos x * cos y) / (cos x * sin y)
5. Now, let's compare the two sides of the equation:
We have (sin x * sin y + cos x * cos y) / (cos x * sin y) = (sin x * sin y - cos x * cos y) / (cos x * sin y)
The terms in the numerator and denominator on both sides are equal, so the equation is verified.
Now, let's move on to the second equation:
sin^4x + cos^4x = 1 - 2cos^2x + 2cos^4
1. Start by using the identity sin^2x + cos^2x = 1:
Substitute this identity into the equation:
(sin^2x)^2 + cos^4x = 1 - 2cos^2x + 2cos^4
2. Simplify each term:
(sin^2x * sin^2x) + cos^4x = 1 - 2cos^2x + 2cos^4
3. Expand the terms:
(sin^4x) + cos^4x = 1 - 2cos^2x + 2cos^4
The terms on both sides of the equation are equal, so the second equation is also verified.
In both cases, we went through the process of simplifying and comparing each side of the equation to determine if they are equal.