Question

please solve and take 50 points ​

Answer 1

To Prove:

$\sf \Longrightarrow sin \theta(1 + tan\theta) + cos \theta(1 + cot\theta) = sec \theta + cosec\theta$

Taking the LHS we get:

$\sf \Longrightarrow sin \theta(1 + tan\theta) + cos \theta(1 + cot\theta)$

We're asked to prove that this expression is equal to secθ + cosecθ.

We know that we can express this expression in the form of secθ and cosecθ if we express the whole expression in terms of sinθ and cosθ.

(Because 1/sinθ = cosecθ and 1/cosθ = secθ)

Using the below ratio we'll get:

⇒ tanθ = sinθ/cosθ

⇒ tanθ = cosθ/sinθ

$\sf \Longrightarrow sin \theta\Bigg[1 + \dfrac{sin \theta}{cos \theta}\Bigg] + cos \theta\Bigg[1 + \dfrac{cos \theta}{sin \theta}\Bigg]$

$\sf \Longrightarrow sin \theta\Bigg[\dfrac{cos \theta + sin \theta}{cos \theta}\Bigg] + cos \theta\Bigg[\dfrac{sin \theta + cos \theta}{sin \theta}\Bigg]$

$\sf \Longrightarrow \dfrac{sin \theta\big[cos \theta + sin \theta\big]}{cos \theta} + \dfrac{cos\theta\big[sin \theta + cos \theta\big]}{sin \theta}$

Taking LCM we get;

$\sf \Longrightarrow \dfrac{sin^2\theta\big[cos \theta + sin \theta\big] + cos^2\theta\big[sin \theta + cos \theta\big]}{sin \theta cos \theta}$

Take [cosθ + sinθ] outside the bracket since it's a common factor.

$\sf \Longrightarrow \dfrac{cos\theta + sin\theta \Big[sin^2\theta + cos^2\theta\Big]}{sin \theta cos \theta}$

We know that:

⇒ sin²θ + cos²θ = 1

$\sf \Longrightarrow \dfrac{cos\theta + sin\theta \Big[1\Big]}{sin \theta cos \theta}$

$\sf \Longrightarrow \dfrac{cos\theta + sin\theta}{sin \theta cos \theta}$

Splitting the numerator we get:

$\sf \Longrightarrow \dfrac{cos\theta}{sin \theta \ cos \theta} + \dfrac{sin\theta}{sin \theta \ cos \theta}$

$\sf \Longrightarrow \dfrac{1}{sin \theta} + \dfrac{1}{cos \theta}$

We know that:

⇒ 1/sinθ = cosecθ

⇒ 1/cosθ = secθ

Substitute these values in the previous line.

$\sf \Longrightarrow cosec \theta + sec \theta$

LHS = RHS

Hence proved.

Answer 2

Our world has always been faced with the problem of discrimination. It is one of the most discussed topics nowadays and throughout history. In all countries there is most likely at least one type of discrimination that affects different groups of people. The definition of discrimination is the denial of opportunity or equal rights for a specific group of people that may be differentiated by things such as their religion, color of skin, or gender. Discrimination can be confused with other terms such as prejudice and stereotype. The world we live in has been struggling with this sensitive subject for as long as we have recorded. Stereotypes are images held in our minds in regards to certain racial or cultural groups, without consideration of whether the images held are true or false. Stemming from stereotypes is prejudice. The prejudicial attitude occurs when we prejudge a person, good or bad, on the basis that the stereotypes associated with the person or group being prejudged are true. Discrimination is the combination of the terms mentioned above, but involves actually acting out with unfair treatment, directing the action towards the person or group. Prejudice and discrimination do not just occur racially, but it is found among gender, religion, culture, and geographical background. Remember that prejudice is a result of attitude and discrimination is a result of action.

Many people believe discrimination has made big steps forward, but has it really? If it has, why do people still get turned down, receive hate mail, or get ridiculed simply because they differ from each other. I guess these are questions we must ask ourselves. I guess you could also ask yourself if you have ever called anybody a name, looked at them different or judged them when you did not even know them or understand them. You may be thinking “That is not discrimination,” but, in fact, it is.

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