{1/(sec x - 1)} - {1/(sec x +1)} = 2 cot^2 x
solve using identities?
Hey, I did this for you long ago.
Twice in fact
copy again
sin a cos b + cos a sin b = sin (a+b)
so sin (7t)
1/(1/cos x -1) - 1/(cos x +1)
cos x/(1 - cos x) - cos x/(1+cos x)
[cos x(1+cos x) - cos x(1-cos x)]/(1-cos^2 x)
[ cos x + cos^2x - cos x +cos^2 x]/sin^2x
2 cos^2 x/sin^2 x
2 cot^2 x
ohh, i didn't realize! thankyou :)
Please check before repeating posts. Some other teacher might have done it all over again.
To solve the given equation using trigonometric identities, we need to manipulate the equation and simplify it until we reach an identity that we can solve.
Let's start by simplifying the left side of the equation:
{1/(sec x - 1)} - {1/(sec x +1)}
To simplify this expression, we need to find a common denominator. The common denominator here is (sec x - 1)(sec x +1):
{(sec x + 1) - (sec x - 1)} / (sec x - 1)(sec x + 1)
Simplifying the numerator gives:
(sec x + 1 - sec x + 1) / (sec x - 1)(sec x + 1)
2 / (sec x - 1)(sec x + 1)
Now, let's simplify the right side of the equation:
2 cot^2 x
Since cotangent is the reciprocal of tangent, we can rewrite cot^2 x as 1/(tan^2 x):
2 / (1/(tan^2 x))
To divide by a fraction, we can multiply by the reciprocal, so this becomes:
2 * (tan^2 x / 1)
2 tan^2 x
Now, our equation becomes:
2 / (sec x - 1)(sec x + 1) = 2 tan^2 x
Next, we notice that 2 is a common factor on both sides, so we can cancel it out:
1 / (sec x - 1)(sec x + 1) = tan^2 x
Now, we can use an identity to simplify the left side of the equation:
1 / (sec^2 x - 1) = tan^2 x
We know that sec^2 x - 1 is equal to tan^2 x (from the Pythagorean identity), so we can substitute tan^2 x for sec^2 x - 1:
1 / tan^2 x = tan^2 x
Now, we multiply both sides of the equation by tan^2 x to get:
1 = tan^4 x
Taking the square root of both sides gives:
1 = tan^2 x
The solutions to this equation are the angles whose tangent is equal to 1. The tangent of an angle is equal to 1 when the angle is equal to π/4 (45 degrees) or 5π/4 (225 degrees).
Therefore, the solutions to the given equation are x = π/4 + nπ and x = 5π/4 + nπ, where n is an integer.