i have a few questions
cosx + cosx =2secx
----- -----
1+sinx 1-sinx
cos(x-B)-cos(x-B) = 2sinxsinB
csc2x= 1 secx cscx
---
2
cotx= cos5x+cos3x
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sin 5x-sin 3x
for the first one:
LS = cosx/(1+sinx) + cosx/(1-sinx)
getting a common denominator of (1+sinx)(1-sinx)
= [cosx(1-sinx) + cosx(1+sinx)]/(1+sinx)(1-sinx)
= 2cosx/(1-sin^2)
= 2cosx/cos^2x
= 2/cosx
= 2secx
= RS
for cos(x-B)-cos(x-B) = 2sinxsinB
use the formula cos(A-B) = cosAcosB + sinAsinB on the Left Side, it comes apart very nicely after that
For the last two, try changing all trig ratios into sines and cosines.
Show me what you get
for cos(x-B)-cos(x-B) = 2sinxsinB
you must have a typo, the LS is zero the way it stands
I am sure you meant
cos(x-B)-cos(x+B) = 2sinxsinB
i don't understand how to use the formula in the second problem. what would i plug into A and B?
cos(x-B)-cos(x+B) = 2sinxsinB
LS
= cosxcosB + sinxsinB - (cosxcosB - sinxsinB)
=2sinxsinB
= RS
Sure, I can help you with these questions.
1. cosx + cosx = 2secx/(1+sinx)/(1-sinx)
To simplify this equation, we need to use trigonometric identities. The identity we can use here is: secx = 1/cosx.
Rewriting the equation using this identity, we get: 2/cosx = 2secx.
Now, we can substitute this back into the original equation:
2secx/(1+sinx)/(1-sinx) = 2/(cosx(1+sinx))/(1-sinx)
To simplify this further, we can multiply the numerator and denominator by (1+sinx)(1-sinx):
2(1+sinx)(1-sinx)/(cosx(1+sinx))/(1-sinx)
Finally, cancel out the common factors in the numerator and denominator, giving us the simplified expression: 2/(cosx/cosx) = 2
2. cos(x-B)-cos(x-B) = 2sinxsinB
To solve this equation, we can use the trigonometric identity: cos(A-B) = cosAcosB + sinAsinB.
Substituting this identity into the equation, we get:
cos(x)cos(-B) + sin(x)sin(-B) - (cos(x)cos(-B) - sin(x)sin(-B)) = 2sinxsinB
Simplifying the equation, we have:
-2sin(x)sin(-B) = 2sinxsinB
Using the identity: sin(-x) = -sinx, we can rewrite the equation as:
2sin(x)sin(B) = 2sinxsinB
Now, we can cancel out the common factors on both sides of the equation, resulting in the simplified expression: sin(x)sin(B) = sinxsinB
3. csc2x = 1/secx cscx/2
To solve this equation, we need to manipulate the left-hand side using trigonometric identities.
The identity we can use here is: cscx = 1/sinx.
Substituting this into the equation, we get:
1/sin2x = 1/secx * 1/sinx / 2
Now, let's simplify the right-hand side of the equation by converting secx into its reciprocal form using the identity: secx = 1/cosx:
1/sin2x = 1/(1/cosx) * 1/sinx / 2
Simplifying further, we have:
1/sin2x = cosx/sinx * 1/sinx / 2
To combine the fractions, we need to find a common denominator, which is sin2x:
1/sin2x = cosx * 1/ sinx * sin2x / 2
On the right-hand side, we can cancel out the sinx term:
1/sin2x = cosx * 1/ 2
Simplifying the equation, we have the simplified expression: 1/sin2x = cosx/2
4. cotx = (cos5x + cos3x) / (sin5x - sin3x)
To solve this equation, we can use the trigonometric identity: cotx = cosx/sinx.
Substituting this identity into the equation, we get:
cosx/sinx = (cos5x + cos3x) / (sin5x - sin3x)
To simplify the equation, we need to manipulate the right-hand side.
Using the identity: sin(A+B) = sinAcosB + cosAsinB, we can rewrite the equation as:
cosx/sinx = (cos5xcosx + cos3xcosx) / (sin5xcosx - sin3xsinx)
Simplifying further, we have:
cosx/sinx = (cos5xcosx + cos3xcosx) / (cosxsin5x - sinxsin3x)
Now, cancel out the common factor of cosx in the numerator and sinx in the denominator:
1/sinx = (cos5x + cos3x) / (cos5x - sin3x)
Cross-multiplying, we get:
sinx = (cos5x + cos3x) * (cos5x - sin3x)
Expanding the numerator and simplifying, we have the final expression for this equation: sinx = cos²5x - sin3x * sin5x + cos3x * cos5x