Prove that (cosy/1+siny) + (1+siny/cosy) = 2secy
Left Side
= cosy/(1+siny) + (1+siny)/cosy , notice the proper brackets !!
= (cos^ y + 1 + 2siny + sin^2 y)/[cosy(1+siny)]
= (2 + 2siny)/[cosy(1+siny)]
= 2(1+siny)/[cosy(1+siny)]
= 2/cosy
= 2secy
= Right Side
To prove that (cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y)) = 2sec(y), we will simplify the left side expression step-by-step.
Step 1: Let's start by finding the common denominator between the two fractions on the left side of the equation, which is cos(y) * (1 + sin(y)).
(cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y)) = (cos(y) * cos(y))/ (cos(y) * (1+sin(y))) + ((1+sin(y)) * (1+sin(y))) / (cos(y) * (1+sin(y)))
Step 2: Simplify the fractions:
(cos(y) * cos(y)) / (cos(y) * (1+sin(y))) = cos^2(y) / (cos(y) + cos(y) * sin(y))
((1+sin(y)) * (1+sin(y))) / (cos(y) * (1+sin(y))) = (1 + 2*sin(y) + sin^2(y)) / (cos(y) + cos(y) * sin(y))
Step 3: Combine the fractions:
(cos^2(y) / (cos(y) + cos(y) * sin(y))) + ((1 + 2*sin(y) + sin^2(y)) / (cos(y) + cos(y) * sin(y)))
Step 4: Combine the numerators and keep the common denominator:
(cos^2(y) + 1 + 2*sin(y) + sin^2(y)) / (cos(y) + cos(y) * sin(y))
Step 5: Simplify the numerator:
(cos^2(y) + sin^2(y) + 1 + 2*sin(y)) / (cos(y) + cos(y) * sin(y))
Step 6: Apply the trigonometric identity sin^2(y) + cos^2(y) = 1:
(1 + 1 + 2*sin(y)) / (cos(y) + cos(y) * sin(y))
Step 7: Combine like terms:
(2 + 2*sin(y)) / (cos(y) + cos(y) * sin(y))
Step 8: Factor out 2:
2 * (1 + sin(y)) / (cos(y) + cos(y) * sin(y))
Step 9: Apply the trigonometric identity 1 + sin(y) = sec(y):
2 * sec(y) / (cos(y) + cos(y) * sin(y))
Step 10: Factor out cos(y) in the denominator:
2 * sec(y) / (cos(y) * (1 + sin(y)))
Step 11: Apply the trigonometric identity 1 + sin(y) = sec(y) again:
2 * sec(y) / (cos(y) * sec(y))
Step 12: Simplify:
2 * sec(y) / sec(y) = 2
Therefore, (cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y)) = 2sec(y) has been proven.
To prove that (cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y)) = 2sec(y), we need to work on the left-hand side of the equation and simplify it until it becomes equal to the right-hand side (2sec(y)).
Starting with the left-hand side:
(cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y))
To combine the two fractions, we need a common denominator. In this case, the common denominator is (1 + sin(y)) * cos(y):
(cos(y) * cos(y) + (1 + sin(y)) * (1 + sin(y))) / (cos(y) * (1 + sin(y)))
Expanding the parentheses:
(cos^2(y) + 2*sin(y) + sin^2(y) + 1) / (cos(y) * (1 + sin(y)))
Simplifying the numerator by combining like terms (cos^2(y) + sin^2(y) = 1):
(1 + 2*sin(y) + 1) / (cos(y) * (1 + sin(y)))
Simplifying further:
(2 + 2*sin(y)) / (cos(y) * (1 + sin(y)))
Factoring out 2:
2 * (1 + sin(y)) / (cos(y) * (1 + sin(y)))
Now, canceling out the common factor (1 + sin(y)) in the numerator and denominator:
2 / cos(y)
Finally, using the definition of sec(y) (sec(y) = 1/cos(y)):
2sec(y)
Therefore, we have proven that (cos(y)/(1+sin(y))) + ((1+sin(y))/cos(y)) = 2sec(y).