Hi, I need a bit of help with verifying identities. The problems are as follows:
sinx+cosx/cotx+1=sinx
sin2x/sin - cos2x/cosx=secx
1-tan^2θ/1+tan^2θ=cos2θ
sin x + cos x / cos x/sin x +1 = ?
sin x + sin x + 1 = sin x no way
maybe you mean
(sinx+cosx)/ (cotx+1)
(sin x + cos x) / (cos x/sin x + sin x/sin x)
sin x ( sin x + cos x) / (cos x + sin x)
= sin x yes
Parentheses are VITAL !!!!!
sin2x/sin - cos2x/cosx
= 2sinx cosx/sinx - (1-2sin^2x)/cosx
= 2cosx - secx + 2sin^2x/cosx
= (2cos^2x+2sin^2x)/cosx - secx
see if you can finish it from here
1-tan^2θ/1+tan^2θ
= (1-(sec^2θ - 1)/sec^2θ
= (2 - sec^2θ)/sec^2θ
see if you can finish it off from here.
Sure, I can help you with verifying identities. Let's go through each problem step by step.
1. Verifying the identity: sinx + cosx / cotx + 1 = sinx
To solve this problem, we need to simplify both sides of the equation until they are equal.
Starting with the left side of the equation:
sinx + cosx / cotx + 1
First, we want to simplify the expression cotx + 1. To do that, we can recall the definition of cotangent, which is cosx / sinx. So, replacing cotx with its equivalent, we get:
sinx + cosx / (cosx / sinx) + 1
Next, we simplify the expression inside the parentheses:
sinx + cosx * (sinx / cosx) + 1
The cosx in the numerator cancels out with the cosx in the denominator:
sinx + sinx + 1
Adding the like terms, we get:
2sinx + 1
Now comparing this with the right side of the equation (sinx), we can see that they are not equal. Therefore, the given equation sinx + cosx / cotx + 1 = sinx is not true for all values of x.
2. Verifying the identity: sin2x / sin - cos2x / cosx = secx
For this problem, we will follow a similar approach.
Starting with the left side of the equation:
sin2x / sin - cos2x / cosx
We know that sin2x can be expressed as 2sinxcosx and cos2x can be expressed as 1 - 2sin^2x. Using these trigonometric identities, we can rewrite the equation as:
2sinxcosx / sin - (1 - 2sin^2x) / cosx
First, let's simplify the expression in the denominator, sin - (1 - 2sin^2x):
We can distribute the negative sign, giving us:
sin - 1 + 2sin^2x / cosx
Now, let's combine the terms in the numerator:
(2sin^2x - 1 + sin) / cosx
To further simplify the expression, let's focus on the numerator:
2sin^2x - 1 + sin
Now, we notice that sin = sinx, so we can replace sin with sinx:
2sin^2x - 1 + sinx
Unfortunately, we cannot further simplify the numerator to get it to match with the right side of the equation (secx). Therefore, the given equation sin^2x / sin - cos2x / cosx = secx is not true for all values of x.
3. Verifying the identity: 1 - tan^2θ / 1 + tan^2θ = cos2θ
For this problem, we will again simplify both sides of the equation.
Starting with the left side of the equation:
1 - tan^2θ / 1 + tan^2θ
We know that tan^2θ = sin^2θ / cos^2θ (from the definition of tangent), so we can substitute this into the equation:
1 - (sin^2θ / cos^2θ) / 1 + (sin^2θ / cos^2θ)
Next, let's simplify the expression inside the parentheses:
1 - sin^2θ / cos^2θ + 1
Now, we want to combine the terms in the numerator:
(1 * cos^2θ - sin^2θ) / cos^2θ + 1
Using the identity cos^2θ - sin^2θ = cos2θ, we can rewrite the equation as:
cos2θ / cos^2θ + 1
Next, let's simplify the expression in the denominator, cos^2θ + 1:
Again, we use the identity cos^2θ + sin^2θ = 1:
1 / cos^2θ + 1
Now, we can combine the terms:
cos2θ / (cos^2θ + 1)
Using the identity cos2θ = cos^2θ - sin^2θ, we have:
cos2θ / (cos^2θ - sin^2θ + sin^2θ)
The sin^2θ terms cancel out, leaving us with:
cos2θ / 1
Simplifying further, we get:
cos2θ
Now comparing this with the right side of the equation (cos2θ), we can see that they are equal. Therefore, the given equation 1 - tan^2θ / 1 + tan^2θ = cos2θ is true for all values of θ.
I hope this explanation helps you understand the process of verifying trigonometric identities! If you have any further questions, feel free to ask.