Prove (1+secx)/(tanx+sinx)=cscx
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tan ( x ) = sin ( x ) / cos ( x )
tan ( x ) + sin ( x ) =
sin ( x ) / cos ( x ) + sin ( x ) =
sin ( x ) * [ 1 / cos ( x ) + 1 ] =
sin ( x ) * [ sec ( x ) + 1 ] =
sin ( x ) * [ 1 + sec ( x )]
[ 1 + sec ( x ) ] / [ tan ( x ) + sin ( x ) ] =
[ 1 + sec ( x ) ] / [ sin ( x ) * [ 1 + sec ( x ) ] ] =
1 / sin ( x ) = cosec ( x )
To prove that (1 + sec(x))/(tan(x) + sin(x)) is equal to csc(x), we need to simplify the left-hand side of the equation and transform it into the right-hand side.
Let's begin by simplifying the left-hand side of the equation. First, let's recall the definitions of sec(x), tan(x), and csc(x):
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
csc(x) = 1/sin(x)
Substituting these definitions into the left-hand side of the equation, we have:
(1 + 1/cos(x))/(sin(x)/cos(x) + sin(x))
Next, we will find a common denominator for both terms in the numerator of the left-hand side. The common denominator can be obtained by multiplying the two denominators together, which gives us cos(x) * cos(x) = cos^2(x):
((cos(x) + 1)/cos(x))/(sin(x) + sin(x) * cos(x)/cos(x))
Simplifying the terms in the numerator further, we have:
(cos(x) + 1)/cos(x)
Now, we can rewrite the denominator using the identity csc(x) = 1/sin(x):
(sin(x) + sin(x) * cos(x)/cos(x)) = sin(x) * (1 + cos(x))/cos(x) = sin(x) * sec(x)
Substituting this simplification back into the expression, we get:
(cos(x) + 1)/cos(x) / (sin(x) * sec(x))
Now, we can multiply the numerator and denominator by cos(x) to simplify further:
(cos(x) + 1)/(cos(x) * sin(x) * sec(x))
Using the identity sec(x) = 1/cos(x), we can substitute sec(x) in terms of cos(x) into the expression:
(cos(x) + 1)/(cos(x) * sin(x) * (1/cos(x)))
Simplifying, we have:
(cos(x) + 1)/(sin(x))
Now, we have:
cos(x)/sin(x) + 1/sin(x)
Using the identity csc(x) = 1/sin(x), we can rewrite the expression as:
cot(x) + csc(x)
We know that cot(x) + csc(x) is equal to csc(x), so:
cot(x) + csc(x) = csc(x)
Thus, we have proven that (1 + sec(x))/(tan(x) + sin(x)) is equal to csc(x).