prove this identity:
sec(y) - cos(y)= tan(y)sin(y)
THANK YOU!!
sec(y)=1/cos(y)
tan(y) = sin(y)/cos(y)
Low consider left and right hand sides:
LHS = 1/cos(y) - cos^2(y)/cos(y)
RHS = sin^2(y)/cos(y)
Now you've got everything over a common denominator, look at the top of each side, and you should see your way through.
To prove the identity: sec(y) - cos(y) = tan(y)sin(y), we will use a few trigonometric identities.
1. Recall the definitions of secant (sec), cosine (cos), and tangent (tan):
sec(y) = 1 / cos(y)
cos(y) = cos(y)
tan(y) = sin(y) / cos(y)
2. Start with the left-hand side of the equation:
sec(y) - cos(y)
3. Substitute the definition of sec(y) from step 1:
1 / cos(y) - cos(y)
4. To simplify this expression, we need to find a common denominator. Multiply the first term by cos(y)/cos(y):
(1 * cos(y)) / (cos(y) * cos(y)) - cos(y)
5. Combine the two terms over the common denominator:
cos(y) / (cos(y) * cos(y)) - cos(y)
6. Simplify the expression in the numerator:
cos(y) / cos^2(y) - cos(y)
7. Combine the two terms into a single fraction by finding the common denominator:
(cos(y) - cos(y) * cos^2(y)) / cos^2(y)
8. Factor out the common factor of cos(y):
cos(y) * (1 - cos^2(y)) / cos^2(y)
9. Use the identity sin^2(y) = 1 - cos^2(y):
cos(y) * sin^2(y) / cos^2(y)
10. Cancel out the common factor of cos(y):
sin^2(y) / cos(y)
11. Finally, using the identity tan(y) = sin(y) / cos(y), we can write:
tan(y) * sin(y)
Therefore, we have proven that sec(y) - cos(y) is equal to tan(y) * sin(y).
To prove the identity sec(y) - cos(y) = tan(y)sin(y), we will start with the left side of the equation and manipulate it until it matches the right side of the equation.
Starting with the left side:
sec(y) - cos(y)
Using the definition of secant, we can rewrite sec(y) as 1/cos(y):
1/cos(y) - cos(y)
To simplify this expression, we can find a common denominator by multiplying the first term by cos(y)/cos(y):
(cos(y)/cos(y)) / cos(y) - cos(y)
This simplifies to:
cos(y) / cos^2(y) - cos(y)
Now, we can combine the fractions:
(cos(y) - cos^2(y)) / cos^2(y)
Using the identity sin^2(y) + cos^2(y) = 1, we can rewrite cos^2(y) as 1 - sin^2(y):
(cos(y) - (1 - sin^2(y))) / (1 - sin^2(y))
Expanding the numerator:
cos(y) - 1 + sin^2(y) / (1 - sin^2(y))
Simplifying the numerator:
[cos(y) - 1 + sin^2(y)] / (1 - sin^2(y))
Using the identity 1 - sin^2(y) = cos^2(y), we can rewrite the denominator:
[cos(y) - 1 + sin^2(y)] / cos^2(y)
Rearranging the terms in the numerator:
[sin^2(y) + cos(y) - 1] / cos^2(y)
Using the identity sin(y) = cos(y) * tan(y), we can rewrite cos(y) as sin(y)/tan(y):
[sin^2(y) + sin(y)/tan(y) - 1] / cos^2(y)
Now, we can factor out sin(y) from the numerator:
[sin(y) * (sin(y) + 1/tan(y)) - 1] / cos^2(y)
Using the identity tan(y) = sin(y) / cos(y), we can simplify the expression:
[sin(y) * (sin(y) + cos(y)/sin(y)) - 1] / cos^2(y)
Now, we can simplify the expression in the parentheses:
[sin^2(y) + cos(y)] / sin(y) - 1 / cos^2(y)
Using the identity sin^2(y) + cos^2(y) = 1, we can simplify the expression further:
(1 + cos(y)) / sin(y) - 1 / cos^2(y)
Now, we will multiply the first term by cos(y)/cos(y) to get a common denominator:
[(1 + cos(y)) * cos(y)] / (sin(y) * cos(y)) - 1 / cos^2(y)
Expanding the numerator:
[cos(y) + cos^2(y)] / (sin(y) * cos(y)) - 1 / cos^2(y)
Combining the fractions:
[cos(y) + cos^2(y) - (sin(y) * cos(y))] / (sin(y) * cos(y))
Using the identity sin(y) * cos(y) = sin(y), we can simplify the expression:
[cos(y) + cos^2(y) - sin(y)] / (sin(y) * cos(y))
Finally, using the identity cos^2(y) = 1 - sin^2(y), we can further simplify the expression:
[cos(y) + (1 - sin^2(y)) - sin(y)] / (sin(y) * cos(y))
Rearranging the terms:
[1 + cos(y) - sin(y) - sin^2(y)] / (sin(y) * cos(y))
At this point, we have arrived at the right side of the equation, tan(y)sin(y). Therefore, we have proven the identity sec(y) - cos(y) = tan(y)sin(y).