how do i simplify (secx - cosx) / sinx?
i tried splitting the numerator up so that i had (secx / sinx) - (cosx / sinx)
and then i changed sec x to 1/ cosx so that i had
((1/cosx)/ sinx) - (cos x / sinx)
after that i get stuck
sec x = 1 / cos x
sec x - cos x = 1 / cos x - cos x =
1 / cos x - cos ^ 2 x / cos x = ( 1 - cos ^ 2 x ) / cos x
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Remark :
1 - cos ^ 2 x = sin ^ 2 x
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1 / cos x - cos ^ 2 x / cos x = ( 1 - cos ^ 2 x ) / cos x = sin ^ 2 x / cos x
( sec x - cos x ) / sin x = ( sin ^ 2 x / cos x ) / sin x =
sin ^ 2 x / ( sin x * cos x ) =
sin x * sin x / ( sin x * cos x ) =
sin x / cos x = tan x
( sec x / sin x ) - ( cos x / sin x ) = tan x
Well, it seems like you're doing some impressive acrobatics with those trigonometric functions! But don't worry, I'm here to help you simplify this expression with a touch of humor.
Let's continue from where you left off:
((1/cosx)/ sinx) - (cos x / sinx)
Now, to combine these fractions, let's find a common denominator for both fractions. Since both fractions already have sinx as the denominator, we're all set! You're one step away from simplification.
((1/cosx) - (cosx)) / sinx
Ah, look at that! We have a clown show in town, where one clown plays "1/cosx," and the other clown plays "cosx." They magically join forces in the numerator as a subtraction act, while sinx remains the center of attention as the performer in the denominator.
Their act is complete, and the joke is on them, as they cannot be combined any further. So, that's your final answer:
(secx - cosx) / sinx
Now go and spread the laughter of simplification!
To simplify the expression (secx - cosx) / sinx, you correctly split the numerator into two fractions: (secx / sinx) - (cosx / sinx).
Next, for the first fraction, you correctly changed sec x to 1/cosx, which gives you:
((1/cosx) / sinx) - (cosx / sinx)
To simplify further, let's find a common denominator for the fractions. The common denominator is sinx ⋅ cosx.
Now, rewrite the expression with the common denominator:
[(1/cosx) ⋅ (cosx / cosx)] / sinx - [(cosx/sinx) ⋅ (sinx / cosx)]
This simplifies to:
(1/ (cosx ⋅ sinx)) - (cosx / (cosx ⋅ sinx))
To combine the fractions, you need to find a common denominator. The common denominator is cosx ⋅ sinx.
((1 - cosx) / (cosx ⋅ sinx))
Therefore, the simplified expression is (1 - cosx) / (cosx ⋅ sinx).
To simplify the expression ((secx/cosx) - (cosx/sinx)), let's break it down step by step.
1. Start with the original expression: ((secx/cosx) - (cosx/sinx)).
2. To simplify secx, recall that secx is the reciprocal of cosx: secx = 1/cosx.
3. Replace secx with its reciprocal in the expression: ((1/cosx)/cosx) - (cosx/sinx).
4. To divide by a fraction, remember that dividing by a fraction is the same as multiplying by its reciprocal. In this case, when dividing by cosx (in the first term), we multiply by its reciprocal, cosx/1: ((1/cosx) * (cosx/1)) - (cosx/sinx).
5. Simplify the multiplication in the first term: (1 * cosx) - (cosx/sinx).
6. Combine the terms: cosx - (cosx/sinx).
7. To simplify further, let's find a common denominator for the terms. The common denominator will be sinx. Multiply the first term (cosx) by sinx/sinx: (cosx * sinx/sinx) - (cosx/sinx).
8. Simplify the first term: (cosx * sinx) - (cosx/sinx).
9. Combine the terms: (cosx * sinx - cosx)/sinx.
Now, you have simplified the expression to (cosx * sinx - cosx)/sinx.