(secx-cosx) / (tanx-sinx) = cosecx+cotx
Spent hours to solve this question but failed. Please help. Thank you
Unless I see an obvious identity I can replace, I usually change everything to sines and cosines.
LS = (1/cosx - cosx)/(sinx/cosx - sinx)
= ( 1 - cos^2 x)/cosx )/( (sinx - sinxcosx)/cosx )
= ( 1 - cos^2 x)/cosx )* cosx/( (sinx - sinxcosx)
= (1 - cos^2 x)/(sinx(1 - cosx))
= (1 - cosx)(1 + cosx)/(sinx(1 - cosx))
= (1 + cosx)/sinx
= 1/sinx + cosx/sinx
= cscx + cotx
+ RS
To solve the given equation:
(sec(x) - cos(x)) / (tan(x) - sin(x)) = csc(x) + cot(x)
First, let's simplify the left side of the equation:
(sec(x) - cos(x)) / (tan(x) - sin(x))
The first step is to recall the definitions of sec(x), tan(x), and sin(x):
sec(x) = 1 / cos(x)
tan(x) = sin(x) / cos(x)
Using these definitions, we can rewrite the expression as:
(1 / cos(x) - cos(x)) / (sin(x) / cos(x) - sin(x))
The next step is to simplify each fraction individually:
For the numerator:
1 / cos(x) - cos(x)
Multiplying the first term by cos(x)/cos(x) to get a common denominator, we have:
(1 - cos^2(x)) / cos(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator as:
sin^2(x) / cos(x)
For the denominator:
sin(x) / cos(x) - sin(x)
Multiplying the first term by cos(x)/cos(x) to get a common denominator, we have:
(sin(x) - sin(x) * cos(x)) / cos(x)
Factoring out sin(x), we get:
sin(x) * (1 - cos(x)) / cos(x)
Now, let's substitute these simplified fractions back into the original expression:
(sin^2(x) / cos(x)) / (sin(x) * (1 - cos(x)) / cos(x))
The next step is to divide the fractions by multiplying by the reciprocal of the denominator:
(sin^2(x) / cos(x)) * (cos(x) / (sin(x) * (1 - cos(x))))
Canceling out the common terms, we get:
sin(x) / (sin(x) * (1 - cos(x)))
We can further simplify this expression by canceling out the sin(x) terms:
1 / (1 - cos(x))
Using the identity csc(x) = 1 / sin(x), we can rewrite the expression as:
csc(x) + cot(x)
Therefore, the left side of the equation simplifies to:
csc(x) + cot(x)
which is the same as the right side of the equation.
Thus, we have shown that the given equation holds true for all values of x, except where cos(x) = 1 (since dividing by zero is undefined).
Therefore, the given equation (sec(x) - cos(x)) / (tan(x) - sin(x)) = csc(x) + cot(x) is true for all x, except x = 2πn, where n is an integer.