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).