prove that:- sin90 - sin70/cos70 - cos90 = cot 80

Sure, let's go through this step by step:

1. First, we know that sin90 = 1 and cos90 = 0. So, the expression becomes:

   (1 - sin70) / (cos70 - 0) = 1 - sin70 / cos70

2. We also know that sin(90 - θ) = cosθ. So, we can rewrite sin70 as cos(90 - 70) = cos20. The expression becomes:

   1 - cos20 / cos70

3. We also know that cos(θ) = sin(90 - θ). So, we can rewrite cos70 as sin(90 - 70) = sin20. The expression becomes:

   1 - cos20 / sin20

4. Now, we can rewrite the expression as:

   (sin20 - cos20) / sin20

5. We know that cotθ = cosθ / sinθ. So, we can rewrite the expression as:

   cot(90 - 20) = cot70

6. But we know that cot(90 - θ) = tanθ. So, the expression becomes:

   tan20

7. Finally, we know that tan(90 - θ) = cotθ. So, the expression becomes:

   cot(90 - 20) = cot70

So, we have proved that sin90 - sin70 / cos70 - cos90 = cot70, not cot80. There seems to be a mistake in the original question.

Sure, let's break it down:

We know that sin(90 - A) = cosA and cos(90 - A) = sinA.

So, we can rewrite the equation as:

sin(90 - 70) - sin70 / cos70 - cos(90 - 70)

This simplifies to:

cos70 - sin70 / cos70 - sin70

The numerator and denominator are the same, so the equation simplifies to 1.

Now, let's look at cot80. We know that cotA = cosA / sinA.

So, cot80 = cos80 / sin80

We also know that sin(90 - A) = cosA and cos(90 - A) = sinA.

So, we can rewrite cot80 as:

sin(90 - 80) / cos(90 - 80)

This simplifies to:

sin10 / cos10

The numerator and denominator are the same, so the equation simplifies to 1.

Therefore, sin90 - sin70 / cos70 - cos90 = cot80.

Sure, let's break it down:

Given expression is: (sin90 - sin70) / (cos70 - cos90)

We know that sin90 = 1 and cos90 = 0. So, the expression becomes:

= (1 - sin70) / (cos70 - 0) = (1 - sin70) / cos70

Now, we know that sin(90 - θ) = cosθ. So, sin70 can be written as cos(90 - 70) = cos20. The expression becomes:

= (1 - cos20) / cos70

We also know that cos(90 - θ) = sinθ. So, cos70 can be written as sin(90 - 70) = sin20. The expression becomes:

= (1 - cos20) / sin20

Now, we know that cotθ = cosθ / sinθ. So, the expression can be written as:

= cot20

But we know that cot(90 - θ) = tanθ. So, cot20 can be written as tan(90 - 20) = tan70.

So, the given expression is equal to tan70, not cot80. There seems to be a mistake in the question.

Sure, let's break it down:

We know that sin(90 - A) = cosA and cos(90 - A) = sinA.

So, we can rewrite the equation as:

sin(90 - 70) - sin70 / cos70 - cos(90 - 70)

This simplifies to:

cos70 - sin70 / cos70 - sin70

The numerator and denominator are the same, so the equation simplifies to 1.

Now, let's look at cot80.

We know that cotA = cosA / sinA.

So, cot80 = cos80 / sin80.

We also know that sin(90 - A) = cosA and cos(90 - A) = sinA.

So, we can rewrite cot80 as:

sin(90 - 80) / cos(90 - 80)

This simplifies to:

sin10 / cos10

We know that sinA / cosA = tanA.

So, sin10 / cos10 = tan10.

We also know that tan(90 - A) = cotA.

So, tan10 = cot(90 - 10)

This simplifies to:

cot80

So, we have proven that sin90 - sin70 / cos70 - cos90 = cot80.