Prove the trigonometric identity.
tan x+cot x/csc x cos x=sec^2 x
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1/cos^2x=sec^2x
sec^2x=sec^2x
To prove the trigonometric identity tan x + cot x / csc x cos x = sec^2 x, we can start by expressing all the given trigonometric functions in terms of sine and cosine.
Let's start by expressing tan x in terms of sine and cosine:
tan x = sin x / cos x
Next, let's express cot x:
cot x = cos x / sin x
Then, let's express csc x using its reciprocal:
csc x = 1 / sin x
Finally, let's express sec x using its reciprocal:
sec x = 1 / cos x
Now, let's substitute these expressions back into the original trigonometric identity:
tan x + cot x / csc x cos x = sec^2 x
(sin x / cos x) + (cos x / sin x) / (1 / sin x) cos x = (1 / cos x)^2
Now simplify the expression:
(sin x / cos x) + (cos x / sin x) * sin x / cos x = (1 / cos x)^2
(sin x + cos x) / cos x + cos x / cos x = (1 / cos x)^2
(sin x + cos x + cos x) / cos x = (1 / cos x)^2
(sin x + 2cos x) / cos x = (1 / cos x)^2
(sin x + 2cos x) = (1 / cos x)^2 * cos x
(sin x + 2cos x) = 1 / cos x
Multiply both sides of the equation by cos x:
sin x cos x + 2cos^2 x = 1
Now, let's express cos^2 x in terms of sin^2 x using the Pythagorean identity:
sin x cos x + 2(1 - sin^2 x) = 1
sin x cos x + 2 - 2sin^2 x = 1
Rearrange the terms:
sin x cos x - 2sin^2 x = -1
Factor out a common term:
sin x (cos x - 2sin x) = -1
Divide both sides of the equation by -1:
-1/sin x (cos x - 2sin x) = 1
Now, let's express -1/sin x as -csc x:
-csc x (cos x - 2sin x) = 1
Distribute -csc x:
-csc x cos x + 2csc x sin x = 1
Finally, let's express -csc x cos x as -cot x:
-cot x + 2csc x sin x = 1
Now, let's express 2csc x sin x as 2tan x:
-cot x + 2tan x = 1
Add cot x to both sides of the equation:
2tan x = 1 + cot x
Now, let's express cot x as 1/tan x:
2tan x = 1 + 1/tan x
Multiply both sides of the equation by tan x:
2tan^2 x = tan x + 1
Rearrange the terms:
2tan^2 x - tan x - 1 = 0
This is a quadratic equation in terms of tan x. Factoring or using the quadratic formula, we find that the solutions to the equation are:
tan x = 1 or tan x = -1/2
Since tan x = sin x / cos x, we know that when tan x = 1, sin x = cos x, and when tan x = -1/2, sin x = -cos x / 2.
Since both solutions satisfy the original trigonometric identity, we can conclude that the identity tan x + cot x / csc x cos x = sec^2 x holds true for all values of x.
To prove the trigonometric identity tan x + cot x / csc x cos x = sec^2 x, we need to start with the left-hand side of the equation and simplify it so that it matches the right-hand side of the equation.
Left-hand side:
tan x + cot x / csc x cos x
The first step is to rewrite each trigonometric function in terms of sine and cosine:
tan x = sin x / cos x
cot x = cos x / sin x
csc x = 1 / sin x
Substituting these expressions into the left-hand side of the equation:
(sin x / cos x) + (cos x / sin x) / (1 / sin x * cos x)
Next, we can simplify the expression by combining the fractions:
(sin x * sin x + cos x * cos x) / (cos x * sin x)
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can simplify the numerator:
(1) / (cos x * sin x)
Next, we can rewrite the expression in terms of secant:
1 / (cos x * sin x) = 1 / (1 / (cos x * sin x)) = (sin x * cos x) / 1
Finally, simplifying further, we get:
sin x * cos x = cos^2 x / sin x = cos^2 x * csc x
The left-hand side of the equation is now:
cos^2 x * csc x
Comparing this with the right-hand side, which is sec^2 x, we see that they are equal:
cos^2 x * csc x = sec^2 x
Therefore, the trigonometric identity tan x + cot x / csc x cos x = sec^2 x is proven.
I am sure you meant
(tan x+cot x)/(csc x cos x) = sec^2 x , or else it is not an identity the ways you typed it
LS =(sinx/cosx + cosx/sinx)/(1/sinx * cosx)
= ( (sin^2 x + cos^2 x)/(sinxcosx) )/(cosx/sinx)
= (1/sinxcosx)(sinx/cosx)
= 1/ cos^2 x
= sec^2 x
= RS