Find a numerical value of one trigonometric function of x if
tanx/cotx - secx/cosx = 2/cscx
a) cscx=1
b) sinx=-1/2
c)cscx=-1
d)sinx=1/2
(sinx/cosx)/(cosx/sinx) - (1/cosx)/cosx = 2sinx
(sin^2 x)/(cos^2 x) - 1/cos^2 x = 2sinx
(sin^2 x - 1)/cos^2 x = 2sinx
(sin^2 x - 1)/(1 - sin^2 x) = 2sinx
-1 = 2sinx
sinx = -1/2
To find the numerical value of one trigonometric function of x, we need to simplify the given equation and substitute the given values of other trigonometric functions.
The given equation is:
tanx/cotx - secx/cosx = 2/cscx
First, let's simplify the left side of the equation using the definitions of the trigonometric functions:
tanx/cotx = (sinx/cosx)/(cosx/sinx) = sin^2(x)/cos^2(x)
secx/cosx = 1/cosx
2/cscx = 2/sinx
Now we substitute these simplified forms back into the equation:
sin^2(x)/cos^2(x) - 1/cosx = 2/sinx
To simplify further, we'll use the identity sin^2(x) = 1 - cos^2(x):
(1 - cos^2(x))/cos^2(x) - 1/cosx = 2/sinx
Now, let's get rid of the fractions by multiplying through by the common denominator cos^2(x):
(1 - cos^2(x)) - cosx = 2cosx
Expanding and rearranging the terms:
1 - cos^2(x) - cosx = 2cosx
1 - cosx - cos^2(x) = 2cosx
Moving all terms to one side:
cos^2(x) + 3cosx - 1 = 0
Now, to solve this quadratic equation for cos(x), we can apply the quadratic formula:
cosx = (-b ± √(b^2 - 4ac))/(2a)
In this equation, a = 1, b = 3, and c = -1:
cosx = (-3 ± √(3^2 - 4(1)(-1)))/(2(1))
cosx = (-3 ± √(9 + 4))/(2)
cosx = (-3 ± √(13))/(2)
At this point, we have found the possible values for cosx. However, we still need to determine the value of x to find the numerical value of the trigonometric function.
To find the value of x, we can use the inverse cosine function (arccos) to find the angles that have cosx as their cosine value. Since the options provided are in terms of sinx and cscx, we'll convert the cosine values to sine values using the identity sin^2(x) + cos^2(x) = 1.
Knowing that sin^2(x) = 1 - cos^2(x), we can rewrite the identity as sin^2(x) = 1 - (1 - sin^2(x)), which simplifies to sin^2(x) = sin^2(x).
Taking the square root on both sides, we get sin(x) = ±√(sin^2(x)) = ±sin(x), which means sin(x) = sin(x).
From this, we know that the values of x are the same as sin^-1(sin(x)).
Now, let's determine which option matches the solutions by calculating the inverse sine of the possible sine values:
a) cscx = 1
To evaluate, we use sin^-1(1), which is equal to π/2 or 90 degrees.
b) sin(x) = -1/2
To evaluate, we use sin^-1(-1/2), which is equal to -π/6 or -30 degrees.
c) cscx = -1
There is no angle that satisfies cscx = -1 since cscx is always positive or zero.
d) sin(x) = 1/2
To evaluate, we use sin^-1(1/2), which is equal to π/6 or 30 degrees.
Based on our calculations, options a) and d) match with the given equation. Thus, the numerical value of one trigonometric function of x is cscx = 1 and sinx = 1/2.
Therefore, the correct answer is d) sinx = 1/2.