Question

For a right angled triangle with integer sides atleast one of its measurements must be an even number. Why?

Answer 1

A right angled triangle follows Pythagoras theorem. means to say if a , b , c are the sides of right angled triangle then, it should be
$a^2=b^2+c^2\\or,b^2=c^2+a^2\\or,c^2=a^2+b^2$

for a² = b² + c² , a is known as hypotenuse.
if we assume a is an odd number then a² must be an odd number .
Let a² = k
similarly, we can assume b is an odd number
then, b² must be an odd number .
Let b² = m

now, k = c² + m
we see k is an odd number is sum of two different number in which one is also an odd number.e.g., m . then c² should be an even number. because sum of two odd number is always even number . hence for getting odd number of sum of two different number in which one is odd then other must be even number.

hence, c² is an even number . so, c must be an even number.

therefore, For a right angled triangle with integer sides atleast one of its measurements must be an even number.

Answer 2

Answer:

Step-by-step explanation:cosθ/(1 - sinθ) + cosθ/(1 + sinθ) = 4

=> {cosθ(1 + sinθ) + cosθ(1 - sinθ)}/(1 - sinθ)(1 + sinθ) = 4

=> {cosθ + cosθ.sinθ + cosθ - cosθ.sinθ}/(1 - sin²θ) = 4

=> 2cosθ/cos²θ = 4 [ we know, sin²x + cos²x = 1 so, (1 - sin²θ) = cos²θ]

=> 2/cosθ = 4

=> cosθ = 1/2 = cos60°

hence, in 0 < θ < 90° , θ = 60°

now, if given equation is not defined.

(1 - sinθ) = 0

in 0 < θ < 90° , sinθ = 1 at 90°

hence, equation is undefined at θ = 90°

[ note : one more case for undefined, (1 + sinθ) = 0 , but in 0 < θ < 90° it's not possible. thars why I didn't mention it above]