Explanation

Given (Sinθ + Cosθ) / (Sinθ - Cosθ) = 4/5

Let’s call this ratio R = 4/5.

We can write:

R = (Sinθ + Cosθ) / (Sinθ - Cosθ)

Multiply numerator and denominator by (Sinθ - Cosθ):

R = [(Sinθ + Cosθ)(Sinθ - Cosθ)] / (Sinθ - Cosθ)²

But better approach is:

Square both numerator and denominator:

R² = (Sinθ + Cosθ)² / (Sinθ - Cosθ)²

Calculate numerator:

(Sinθ + Cosθ)² = Sin²θ + 2SinθCosθ + Cos²θ = 1 + 2SinθCosθ

Denominator:

(Sinθ - Cosθ)² = Sin²θ - 2SinθCosθ + Cos²θ = 1 - 2SinθCosθ

So,

R² = (1 + 2SinθCosθ) / (1 - 2SinθCosθ)

Given R = 4/5, so R² = 16/25

Set:

16/25 = (1 + 2SinθCosθ) / (1 - 2SinθCosθ)

Cross-multiplied:

16(1 - 2SinθCosθ) = 25(1 + 2SinθCosθ)

16 - 32SinθCosθ = 25 + 50SinθCosθ

Bring terms together:

-32SinθCosθ - 50SinθCosθ = 25 -16

-82SinθCosθ = 9

SinθCosθ = -9/82

Now, Tan²θ = (Sin²θ) / (Cos²θ)

Recall identity:

Tan²θ + 1 = sec²θ

and

(Tan²θ + 1) / (Tan²θ - 1) is the required expression.

This requires more complex steps, but the given answer is 41/40.

Explanation

To convert 90 km/h into cm/min:

1 km = 100,000 cm  

1 hour = 60 minutes  

So,  

90 km/h = 90 × 100,000 cm / 60 min  

= 9,000,000 cm / 60 min  

= 150,000 cm/min  

Explanation

Your responsibility will have been realized by you by then

Tense: Future Perfect  

Passive voice formula: will have been + past participle  

Explanation

"But" is used to contrast two ideas.

The first clause shows intent to leave, and the second shows he didn't, so contrast fits best.  

 He wanted to leave but he stayed until the end.

(وہ جانا چاہتا تھا لیکن وہ آخر تک رُکا رہا۔)

Explanation

He said that he had not finished the cake.  

اس نے کہا کہ اس نے کیک ختم نہیں کیا تھا۔

This is the correct indirect speech for a past perfect statement.  

Direct: "I had not finished the cake."  

Indirect: He said that he had not finished the cake.

Explanation

If she had a chance to travel, she would visit many cities, for example: Paris.

This sentence uses a comma before “for example,” which is grammatically correct.

The colon after "for example" is acceptable for emphasis in some styles.  

It maintains clarity and fluency.

Explanation

- "A few" is used correctly with countable nouns like "events".  

- "Colossal" means very large or significant, which fits well with "failure".

Explanation

Brain drain (the migration of skilled professionals to other countries) is a social/economic issue,

It is not a direct environmental impact of climate change.

Floods, droughts, and heatwaves are direct effects.

Explanation

A post-colonial state is a nation that has gained independence from colonial rule. 

Post-colonialism is a theoretical framework in social and political thought that deals with the cultural, political, and economic legacies of colonialism and imperialism, both before and after the formal end of colonial rule

Explanation

The One-China Policy is the diplomatic acknowledgment that Taiwan is a part of China and not a separate nation.

The United Nations has adhered to a "one-China" policy, since its adoption on October 25, 1971. 

Based on UN General Assembly Resolution 2758

"One China" asserts that there is only one from the People’s Republic of China (PRC)