How do I prove these trig identities?

Question

How do I prove these trig identities?

secx-cosx/tanx =sinx

And

1+sinx/cosx+ cox/ 1+sinx=2secx

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2 answers
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Answer 1

MathGuru answered
13 years ago

I will show you the first problem to give you an idea how to approach proving trig identities.

Here are a few basic identities that will help:

sinx/cosx = tanx
secx = 1/cosx
cos^2x + sin^2x = 1

Let's put everything into sine and cosine form and work the left hand side:

(secx - cosx)/tanx = sinx
(1/cosx - cosx)/(sinx/cosx) = sinx
(1/cosx - cosx)(cosx)/sinx = sinx
(cosx/cosx - cos^2x)/sinx = sinx
(1 - cos^2x)/sinx = sinx
sin^2x/sinx = sinx
sinx = sinx

And there you have it!

Answer 2

Explain Bot answered
1 year ago

To prove these trigonometric identities, we will manipulate the equations step by step using the properties and definitions of trigonometric functions.

First, let's prove the identity: sec(x) - cos(x) / tan(x) = sin(x).

Step 1:
We'll start by manipulating the left side of the equation. Recall the definitions of sec(x), tan(x), and cos(x):

sec(x) = 1 / cos(x)
tan(x) = sin(x) / cos(x)

Now let's substitute these definitions into the equation:

sec(x) - cos(x) / tan(x) = (1 / cos(x)) - cos(x) / (sin(x) / cos(x))

Step 2:
Next, let's simplify the expression by finding the common denominator:

sec(x) - cos(x) / tan(x) = (1 - cos^2(x)) / (sin(x) / cos(x))

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can replace 1 - cos^2(x) with sin^2(x):

sec(x) - cos(x) / tan(x) = sin^2(x) / (sin(x) / cos(x))

Step 3:
Now we can simplify further by canceling out sin(x) in the numerator and denominator:

sec(x) - cos(x) / tan(x) = sin(x) / 1

The left side of the equation is now equal to sin(x), proving the identity.

Now, let's move on to the second identity: 1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = 2sec(x).

Step 1:
We'll start by manipulating the left side of the equation. Let's find the common denominator:

1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (cos(x) + sin(x) + cos(x)) / (cos(x)(1 + sin(x)))

Step 2:
Combine like terms in the numerator:

1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2cos(x) + sin(x)) / (cos(x)(1 + sin(x)))

Step 3:
Now, let's simplify the expression further by canceling out the common factor:

1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2 + sin(x) / cos(x)) / (1 + sin(x))

Using the identity tan(x) = sin(x) / cos(x), we can replace sin(x) / cos(x) with tan(x):

1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2 + tan(x)) / (1 + sin(x))

Step 4:
Now, we'll use the Pythagorean identity sec^2(x) = 1 + tan^2(x) to simplify further. Rearranging the equation, we get:

sec^2(x) - tan^2(x) = 1

Dividing both sides by sec^2(x), we have:

1 - tan^2(x) / sec^2(x) = 1 / sec^2(x)

Now, we can substitute these values back into the equation:

(2 + tan(x)) / (1 + sin(x)) = 1 / sec^2(x)

Step 5:
Using the definition of sec(x) as 1 / cos(x), we can rewrite the right side of the equation:

(2 + tan(x)) / (1 + sin(x)) = cos^2(x)

Step 6:
Finally, we can use the reciprocal identity to further simplify the equation:

(2 + tan(x)) / (1 + sin(x)) = 1 / (1 / cos^2(x))

Simplifying the right side:

(2 + tan(x)) / (1 + sin(x)) = 1 / sec(x)

Now, the left side of the equation is equal to the right side, proving the identity.

Remember, when proving trigonometric identities, it's important to have a good understanding of the properties and definitions of trigonometric functions, as well as the relevant trigonometric identities and manipulation techniques like substitution and simplification.