cosecA−sinA)(secA−cosA)=1tanA+cotA

To solve the given equation, we will simplify both sides step by step.

Starting with the left side of the equation:

(cosecA - sinA)(secA - cosA)

First, let's expand the brackets:

cosecA * secA - cosecA * cosA - sinA * secA + sinA * cosA

Next, let's simplify each term:

1/cosA * 1/secA - 1/sinA * cosA - sinA * 1/secA + sinA * cosA

Simplifying further:

1/(cosA * secA) - cosA/sinA - sinA/secA + sinA * cosA

Now, let's simplify the right side of the equation:

1/tanA + cotA

Since cotA is the reciprocal of tanA, we can rewrite the right side as:

1/tanA + 1/tanA

Combining the terms:

2/tanA

Now, let's compare the simplified left side and right side of the equation:

1/(cosA * secA) - cosA/sinA - sinA/secA + sinA * cosA = 2/tanA

To further simplify, let's convert all trigonometric functions to sine and cosine:

1/(cosA * (1/cosA)) - (cosA/sinA) - (sinA/(1/cosA)) + sinA * cosA = 2/(sinA/cosA)

Simplifying:

1 - cosA/sinA - sinA/cosA + sinA * cosA = 2cosA/sinA

Now, let's multiply both sides by sinA to eliminate the denominators:

sinA - cosA - sinA^2 + sinA^2 * cosA = 2cosA

Simplifying further:

sinA - cosA = 2cosA

Now, let's move all terms to one side of the equation:

sinA - cosA - 2cosA = 0

Combining like terms:

sinA - 3cosA = 0

And that is the simplified form of the equation.