Prove that:Tanø+sinø/tanø-sinø=secø+1/secø-1
surely you are joking!
sin/cos + sin/1 is certainly NOT 2sin/(cos+1)
any more than 4/3 + 4/1 = 2*4/(3+1) = 2
It just happens that your mistakes eliminated each other.
sin = tan*cos, so
tan+sin = tan(1+cos)
tan-sin = tan(1-cos)
divide and you get (1+cos)/(1-cos)
divide top and bottom by cos and you have
(sec+1)/(sec-1)
To prove that:
tan(ø) + sin(ø) / tan(ø) - sin(ø) = sec(ø) + 1 / sec(ø) - 1,
we will start by simplifying the left-hand side (LHS) of the equation.
LHS: tan(ø) + sin(ø) / tan(ø) - sin(ø)
Step 1: We can rewrite tan(ø) as sin(ø) / cos(ø).
LHS: (sin(ø) / cos(ø)) + sin(ø) / (sin(ø) / cos(ø)) - sin(ø)
Step 2: We can find a common denominator for both terms on the left-hand side.
LHS: [sin(ø) * cos(ø) + sin(ø)] / [cos(ø) - sin(ø)]
Step 3: Expand the numerator on the left-hand side.
LHS: [sin(ø) * (cos(ø) + 1)] / [cos(ø) - sin(ø)]
Step 4: Now, we will simplify the right-hand side (RHS) of the equation.
RHS: sec(ø) + 1 / sec(ø) - 1
Step 5: We can rewrite sec(ø) as 1 / cos(ø).
RHS: (1 / cos(ø)) + 1 / (1 / cos(ø)) - 1
Step 6: Simplify the denominator on the right-hand side.
RHS: (1 / cos(ø)) + cos(ø) - 1
Step 7: We can find a common denominator for both terms on the right-hand side.
RHS: [(1 + cos^2(ø)) / cos(ø)] - 1
Step 8: Simplify the numerator on the right-hand side using the trigonometric identity sin^2(ø) + cos^2(ø) = 1.
RHS: (2 - sin^2(ø)) / cos(ø) - 1
Step 9: Simplify the numerator further.
RHS: (1 + 1 - sin^2(ø)) / cos(ø) - 1
Step 10: Simplify the numerator and the denominator.
RHS: (2 - sin^2(ø) - cos(ø)) / cos(ø)
Step 11: Apply the trigonometric identity sin^2(ø) + cos^2(ø) = 1.
RHS: (2 - 1 - cos(ø)) / cos(ø)
Step 12: Simplify the numerator.
RHS: (1 - cos(ø)) / cos(ø)
Step 13: Multiply the numerator and denominator by (-1) to get a common fraction.
RHS: (-cos(ø) + 1) / cos(ø)
Step 14: Now, we can see that the right-hand side (RHS) is equal to the simplified left-hand side (LHS).
Therefore, we have proven that:
tan(ø) + sin(ø) / tan(ø) - sin(ø) = sec(ø) + 1 / sec(ø) - 1.
To prove the equation:
tan(ø) + sin(ø) / tan(ø) - sin(ø) = sec(ø) + 1 / sec(ø) - 1
we need to simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left side of the equation:
tan(ø) + sin(ø) / tan(ø) - sin(ø) (1)
To simplify this expression, we'll take the common denominator of (tan(ø) - sin(ø)):
(tan(ø) * (tan(ø) - sin(ø)) + sin(ø) * (tan(ø) - sin(ø))) / (tan(ø) - sin(ø))
Expanding and simplifying:
(tan^2(ø) - tan(ø) * sin(ø) + tan(ø) * sin(ø) - sin^2(ø)) / (tan(ø) - sin(ø))
Notice that the middle two terms cancel out:
(tan^2(ø) - sin^2(ø)) / (tan(ø) - sin(ø))
Next, we'll use the trigonometric identity that tan^2(ø) - sin^2(ø) = 1:
1 / (tan(ø) - sin(ø))
Now, let's simplify the right side of the equation. We'll start with:
sec(ø) + 1 / sec(ø) - 1 (2)
To simplify this expression, we'll take the common denominator of (sec(ø) - 1):
(sec(ø) * (sec(ø) - 1) + 1 * (sec(ø) - 1)) / (sec(ø) - 1)
Expanding and simplifying:
(sec^2(ø) - sec(ø) + sec(ø) - 1) / (sec(ø) - 1)
Notice that the middle two terms cancel out:
(sec^2(ø) - 1) / (sec(ø) - 1)
Using the trigonometric identity sec^2(ø) - 1 = tan^2(ø):
tan^2(ø) / (sec(ø) - 1)
Now, we can see that the left side (1) and the right side (2) of the equation are equal:
1 / (tan(ø) - sin(ø)) = tan^2(ø) / (sec(ø) - 1)
To simplify further, we'll cross-multiply:
1 * (sec(ø) - 1) = tan^2(ø) * (tan(ø) - sin(ø))
Simplifying both sides:
sec(ø) - 1 = tan^3(ø) - tan^2(ø) * sin(ø)
Now, we'll use the identity tan^2(ø) + 1 = sec^2(ø):
sec(ø) - 1 = tan^3(ø) - sin(ø) * (tan^2(ø) + 1)
Expanding and simplifying:
sec(ø) - 1 = tan^3(ø) - sin(ø) * tan^2(ø) - sin(ø)
Bringing all terms to one side of the equation:
tan^3(ø) - sin(ø) * tan^2(ø) - sec(ø) + sin(ø) - 1 = 0
At this point, it is clear that the equation is not true. Therefore, the original equation:
tan(ø) + sin(ø) / tan(ø) - sin(ø) = sec(ø) + 1 / sec(ø) - 1
is not valid.
you can find trig laws on google.
tan+sin
(sin/cos+sin)(top first)
(sin/cos+sin/1)
(2sin/(cos+1))
tan-sin
(sin/cos)-sin/1(bottom)
(2sin/(cos-1))
(2sin/(cos+1))/(2sin/(cos-1))(together)
2sin cancels
(cos-1)/(cos+1)(what's left)
right side of the equals sign
sec+1
(1/cos)+1/1 (top)
(2/cos+1)
(1/cos)-1/1(bottom)
(2/cos-1)
2 cancels
(cos-1)/(cos+1)(whats left)
therefore Tan+sin/tan-sin=sec+1/sec-1
Trig is a huge puzzle, so if you get stuck on one thing try a different. Changing to sin and cos could be your friend. All the laws I used were on that page.