# How do you prove #cos(x) / (1-sin(x)) = sec(x) + tan(x)#?

Please refer to the Explanation.

Have fun with math!

By signing up, you agree to our Terms of Service and Privacy Policy

To prove cos(x) / (1 - sin(x)) = sec(x) + tan(x), we start with the left side of the equation:

cos(x) / (1 - sin(x))

We can multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + sin(x)). This process is often used to simplify expressions involving trigonometric functions.

(cos(x) / (1 - sin(x))) * ((1 + sin(x)) / (1 + sin(x)))

Expanding the numerator and denominator:

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

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:

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

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

= cos(x) * sec(x)

= sec(x) * cos(x)

= sec(x)

Therefore, cos(x) / (1 - sin(x)) simplifies to sec(x).

Additionally, we know that tan(x) = sin(x) / cos(x). So, adding sec(x) and tan(x) together gives sec(x) + tan(x).

Hence, cos(x) / (1 - sin(x)) = sec(x) + tan(x).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you solve #sin(x) = cos(x)#?
- How to use fundamental identities to simplify 9sinβtanβ+9cosβ ?
- How do you use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle #pi/8#?
- How do you solve #1 - 2(sinx)^2 = cosx, 0 <= x <= 360#. Solve for #x#?
- How do you verify the identity #1/(1-sintheta)+1/(1+sintheta)=2sec^2theta#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7