Question

During the middle ages, Europe made remarkable progress in education. The expansion

of basic education, universities, Greco-Roman knowledge and Islamic knowledge were

the features of the Medieval Age. Complete the table based on this.

Progress in the field of trade and culture in the medieval world led to the beginning of the

modern age. Do you agree with this statement? Substantiate your arguments.

I

The main subjects

offered in Gurukula

system in India.

• Languages

•

•

The major subjects

taught in the medieval

schools.

• Grammar

•

•

The main subjects taught

in universities

• Philosophy

•

•

Medieval Education​

Answer 1

Answer:

Equações D

iferenciais de Primeir

a Ordem Exatas

(y + 2xy³) dx + (1 + 3x²y² + x) dy = 0

Answer 2

Correct Question :-

Prove that :- $\mathsf{\;\dfrac{1 - cos\theta}{1 + cos\theta}}$=$\mathsf{(cosec\theta - cot\theta)^2}$

To Prove :-

$\mathsf{\;\dfrac{1 - cos\theta}{1 + cos\theta}}$=$\mathsf{(cosec\theta - cot\theta)^2}$

Proof :-

$\mathsf\pink{{ \: \: \: \: \: \: \: \: \: \:\:L.H.S :\;\dfrac{1 - cos\theta}{1 + cos\theta}}}$

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies \dfrac{(1 - cos\theta)(1 - cos\theta)}{(1 + cos\theta)(1 - cos\theta)}}$

 We know that : (a + b)(a - b) = a² - b²

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies (1 - cos\theta)(1 + cos\theta) = 1 - cos^2\theta}$

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies \dfrac{(1 - cos\theta)^2}{1 - cos^2\theta}}$

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{1 - cos^2\theta = sin^2\theta}}}$

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies \dfrac{(1 - cos\theta)^2}{sin^2\theta}}$

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies \bigg[\dfrac{1 - cos\theta}{sin\theta}\bigg]^2}$

$\mathsf{ \: \: \: \: \: \: \: \: \: \:\::\implies \bigg[\dfrac{1}{sin\theta} - \dfrac{cos\theta}{sin\theta}\bigg]^2}$

We know that :

$\;\;\boxed{\mathsf{\dfrac{1}{sin\theta} = cosec\theta}}$

$\;\;\boxed{\mathsf{\dfrac{cos\theta}{sin\theta} = cot\theta}}$

$\mathsf\pink{{ \: \: \: \: \: \: \: \: \: \:\::\implies (cosec\theta - cot\theta)^2}}$

Hence (Proved..!)