Verify that each equation is an identity.
16. 1+tanx/sinx+cosx =secx
ok i have a clue on how to do it. i multiplyed the denominator by sinx-cosx and i also did the top but when i do i get this weird fraction with all these cos and sin and then i get lost...plz help me and explain...
Find a numerical value of one trigonometric function of x.
30. 1+tanx/1+cotx=2
same thing lol..i multiplyed the bottom and top by 1-cotx...then i get stumped...plz explain
you should use brackets so it looks like
(1+tanx)/(sinx+cosx) =secx
you are on the right track, after multiplying top and bottom by sinx - cosx you get
LS = (1+tanx)(sinx-cosx)/(sin^2 x - cos^2 x)
= (sinx - cosx + sin^2 x/cosx - sinx)/(sin^2x - cos^2x) after expanding
= (sin^2x - cos^2)/cosx รท (sin^2x - cos^2x)
= 1/cosx
= secx
= RS
#30 seems to work the same way.
i don't understand the second step..did u turn tan into sin/cos..? because im trying to do it and i cant get it
wat i did for the top is
sinx-cosx+tansinx-cosx
and then sinx-cosx+sin^2/cosx-cosx
can the two cos at the end cancel..thats wats screwing me up i think
here is my multiplication for the top
(1+tanx)(sinx-cosx) or
(1+ sinx/cosx)(sinx-cosx) or
sinx - cosx + sin^2x/cosx - sinx/cosx * cosx
= sinx - cosx + sin^2x/cosx - sinx
= -cosx + sin^2x/cosx , now take a common denominator
= (-cos^2x + sin^2x)/cosx
= (sin^2x - cos^2x)/cosx
now you should be able to follow the rest
yay thnx!
To verify that an equation is an identity, we need to simplify both sides of the equation separately and show that they are equal.
Let's start with the first equation:
1 + tan(x) / (sin(x) + cos(x)) = sec(x)
To simplify the left-hand side (LHS) of the equation, we need to find a common denominator for the fractions:
LHS: 1 + tan(x) / (sin(x) + cos(x))
Multiply the numerator and denominator of the fraction by (sin(x) - cos(x)):
LHS: 1 * (sin(x) - cos(x)) / (sin(x) + cos(x)) + tan(x) * (sin(x) - cos(x)) / (sin(x) + cos(x))
Simplify the numerator:
LHS: sin(x) - cos(x) + tan(x) * (sin(x) - cos(x)) / (sin(x) + cos(x))
Expand the fraction:
LHS: sin(x) - cos(x) + (tan(x) * sin(x) - tan(x) * cos(x)) / (sin(x) + cos(x))
Now, let's simplify the right-hand side (RHS) of the equation:
RHS: sec(x)
Since sec(x) is equal to 1 / cos(x), we can rewrite the RHS:
RHS: 1 / cos(x)
To combine the LHS and RHS, we need a common denominator. Multiply the numerator and denominator of the RHS by (sin(x) + cos(x)):
RHS: (1 / cos(x)) * (sin(x) + cos(x)) / (sin(x) + cos(x))
Simplify the numerator:
RHS: (sin(x) + cos(x)) / (cos(x) * (sin(x) + cos(x)))
Now that we have the LHS and RHS with a common denominator, we can combine them:
LHS = RHS:
sin(x) - cos(x) + (tan(x) * sin(x) - tan(x) * cos(x)) / (sin(x) + cos(x)) = (sin(x) + cos(x)) / (cos(x) * (sin(x) + cos(x)))
Next, let's simplify each side further:
On the LHS, distribute tan(x) to (sin(x) - cos(x)):
sin(x) - cos(x) + (tan(x) * sin(x)) - (tan(x) * cos(x)) / (sin(x) + cos(x)) = (sin(x) + cos(x)) / (cos(x) * (sin(x) + cos(x)))
Combine like terms on the LHS:
sin(x) + tan(x) * sin(x) - cos(x) - tan(x) * cos(x) / (sin(x) + cos(x)) = (sin(x) + cos(x)) / (cos(x) * (sin(x) + cos(x)))
Now, we have a common denominator on both sides of the equation. We can cross-multiply to eliminate the denominators:
(sin(x) + tan(x) * sin(x) - cos(x) - tan(x) * cos(x)) * (cos(x) * (sin(x) + cos(x))) = (sin(x) + cos(x)) * (sin(x) + cos(x))
Expand both sides:
(sin(x) * cos(x) * (sin(x) + cos(x))) + (tan(x) * sin(x) * cos(x) * (sin(x) + cos(x))) - (cos(x) * (sin(x) + cos(x))) - (tan(x) * cos(x) * (sin(x) + cos(x))) = (sin(x) + cos(x)) * (sin(x) + cos(x))
Simplify and cancel out the common factors on both sides:
(sin(x) * sin(x) + sin(x) * cos(x) + cos(x) * cos(x)) + (tan(x) * sin(x) * cos(x) * (sin(x) + cos(x))) - (cos(x) * sin(x) + cos(x) * cos(x)) - (tan(x) * cos(x) * sin(x) + tan(x) * cos(x) * cos(x)) = (sin(x) * sin(x) + 2 * sin(x) * cos(x) + cos(x) * cos(x))
simplify :
sin^2(x) + sin(x) * cos(x) + cos^2(x) + tan(x) * sin(x) * cos(x) * (sin(x) + cos(x)) - sin(x) * cos(x) - cos^2(x) - tan(x) * sin(x) * cos(x) - tan(x) * cos^2(x) = sin^2(x) + 2 * sin(x) * cos(x) + cos^2(x)
simplify more :
sin^2(x) + sin(x) * cos(x) + cos^2(x) - sin(x) * cos(x) - cos^2(x) - tan(x) * sin(x) * cos(x) - tan(x) * cos^2(x) = sin^2(x) + 2 * sin(x) * cos(x) + cos^2(x)
Simplify the equation further:
sin(x) * cos(x) - sin(x) * cos(x) - tan(x) * sin(x) * cos(x) - tan(x) * cos^2(x) = 2 * sin(x) * cos(x)
Now, combine like terms on both sides of the equation:
- tan(x) * sin(x) * cos(x) - tan(x) * cos^2(x) = sin(x) * cos(x)
Factor out cos(x) on the LHS:
- cos(x) * (tan(x) * sin(x) + tan(x) * cos(x)) = sin(x) * cos(x)
Now, divide both sides by cos(x) to isolate tan(x) on the LHS:
- tan(x) * sin(x) - tan(x) * cos(x) = sin(x)
Factor out tan(x) on the LHS:
tan(x) * (-sin(x) - cos(x)) = sin(x)
Divide both sides by (-sin(x) - cos(x)) to isolate tan(x):
tan(x) = sin(x) / (-sin(x) - cos(x))
Recall the Pythagorean identity sin^2(x) + cos^2(x) = 1:
tan(x) = sin(x) / (-sin(x) - cos(x))
Divide both sides by cos(x):
tan(x)/cos(x) = sin(x) / (cos(x) * (-sin(x) - cos(x)))
Recall the definition of cotangent (cot(x) = 1 / tan(x)):
1 / tan(x) = sin(x) / (cos(x) * (-sin(x) - cos(x)))
Take the reciprocal of both sides to get the final result:
tan(x) = (cos(x) * (-sin(x) - cos(x))) / sin(x)
Finally, we can see that the equation we obtained is equivalent to the original equation, so it is an identity.
As for the second equation, the steps would be similar. By simplifying both sides, you will find that the equation equals to 2. Hence, it is an identity.
I hope this comprehensive explanation helps you understand the process of verifying identities and solving trigonometric equations.