$\tan A+\tan B + \tan A.\tan B = 1\\\\ \implies \: \tan A + \tan B = 1 - \tan A.\tan B \\ \\ \implies \: \frac{\tan A + \tan B}{ 1 - \tan A.\tan B } = 1\\ \\ \implies \: \tan (A + B) = 1 \\ \\ \implies \: \tan (A + B) =\tan 45 \degree \\ \\ \implies \: A + B = 45 \degree \\ \\ thus \: proved$

8 0

2 years ago

Prove Euler's identity using Euler's formula.<br> e^ix = cos x + i sin x

Korolek [52]

First list all the terms out.

e^ix = 1 + ix/1! + (ix)^2/2! + (ix)^3/3! ...

Then, we can expand them.

e^ix = 1 + ix/1! + i^2x^2/2! + i^3x^3/3!...

Then, we can use the rules of raising i to a power.

e^ix = 1 + ix - x^2/2! - ix^3/3!...

Then, we can sort all the real and imaginary terms.

e^ix = (1 - x^2/2!...) + i(x - x^3/3!...)

We can simplify this.

e^ix = cos x + i sin x

This is Euler's Formula.

What happens if we put in pi?

x = pi

e^i*pi = cos(pi) + i sin(pi)

cos(pi) = -1

i sin(pi) = 0

e^i*pi = -1 OR e^i*pi + 1 = 0

That is Euler's identity.

3 0

3 years ago

A vending machine has 5 colors (white, red, green, blue, and yellow) of gumballs and an equal chance of dispensing each. A secon

babunello [35]

Answer:

1/20

Step-by-step explanation:

6 0

3 years ago