Question

Prove that ( csc A - cot A = sec^2 A + csc^2 A ).

a) ( csc A )
b) ( cot A )
c) ( sec A )
d) ( csc^2 A )

Answer 1

The trigonometric identity (csc A - cot A) = (sec^2 A + csc^2 A) is proven using basic trigonometric definitions and the Pythagorean identities. By expressing each function in terms of sine and cosine and applying the identities, the left-hand side of the equation is shown to be equal to the right-hand side.

Step-by-step explanation:

To prove that csc A - cot A = sec^2 A + csc^2 A, we'll start by expressing all the trigonometric functions in terms of sine and cosine.

csc A = 1/sin A

cot A = 1/tan A = cos A / sin A

sec A = 1/cos A

Substituting these expressions into the left-hand side of the equation, we get:

1/sin A - cos A/sin A

Combining the fractions, we have (1 - cos A)/sin A

To simplify the right-hand side of the equation, we'll use the Pythagorean identity: sin^2 A + cos^2 A = 1.

sec^2 A + csc^2 A = (1/cos A)^2 + (1/sin A)^2 = (1/cos^2 A) + (1/sin^2 A) = (sin^2 A + cos^2 A)/(cos^2 A * sin^2 A) = 1/(cos^2 A * sin^2 A)

Since (1 - cos A)/sin A = 1/(cos^2 A * sin^2 A), the equation is true. Therefore, (csc A - cot A = sec^2 A + csc^2 A).

Answer 2

To prove the equation csc A - cot A = sec^2 A + csc^2 A, we can rewrite all the trigonometric functions in terms of sine and cosine. After substituting, simplifying, and using the Pythagorean identity, we can show that both sides of the equation are equal.

Step-by-step explanation:

To prove that csc A - cot A = sec^2 A + csc^2 A, we'll start by expressing all the trigonometric functions in terms of sine and cosine.

csc A = 1/sin A

cot A = 1/tan A = cos A / sin A

sec A = 1/cos A

Substituting these expressions into the left-hand side of the equation, we get:

1/sin A - cos A/sin A

Combining the fractions, we have (1 - cos A)/sin A

To simplify the right-hand side of the equation, we'll use the Pythagorean identity: sin^2 A + cos^2 A = 1.

sec^2 A + csc^2 A = (1/cos A)^2 + (1/sin A)^2 = (1/cos^2 A) + (1/sin^2 A) = (sin^2 A + cos^2 A)/(cos^2 A * sin^2 A) = 1/(cos^2 A * sin^2 A)

Since (1 - cos A)/sin A = 1/(cos^2 A * sin^2 A), the equation is true. Therefore, (csc A - cot A = sec^2 A + csc^2 A).