Check your questions is it tan²theta + cot²theta +2 =sec²theta×cosec²theta

Proof:

To prove that tan^2(theta) * cot^2(theta) = cos^2(theta) * cosec^2(theta), we will first express each term in terms of sine and cosine functions.

Step 1: Expressing tan^2(theta) and cot^2(theta) in terms of sine and cosine:

Using the trigonometric identity, tan^2(theta) = sec^2(theta) - 1, we can rewrite tan^2(theta) as:

tan^2(theta) = sec^2(theta) - 1

Similarly, using the identity, cot^2(theta) = cosec^2(theta) - 1, we can rewrite cot^2(theta) as:

cot^2(theta) = cosec^2(theta) - 1

Step 2: Expressing sec^2(theta) and cosec^2(theta) in terms of sine and cosine:

Using the identity, sec^2(theta) = 1 + tan^2(theta), we can express sec^2(theta) as:

sec^2(theta) = 1 + tan^2(theta)

Similarly, using the identity, cosec^2(theta) = 1 + cot^2(theta), we can express cosec^2(theta) as:

cosec^2(theta) = 1 + cot^2(theta)

Step 3: Substituting the expressions for sec^2(theta) and cosec^2(theta) in terms of tan^2(theta) and cot^2(theta):

Substituting the expressions for sec^2(theta) and cosec^2(theta) obtained in Step 2 into the equation from Step 1, we have:

tan^2(theta) * cot^2(theta) = (1 + tan^2(theta)) * (1 + cot^2(theta)) - 1

Expanding the right side of the equation:

tan^2(theta) * cot^2(theta) = 1 + tan^2(theta) + cot^2(theta) + tan^2(theta) * cot^2(theta) - 1

Simplifying the equation:

tan^2(theta) * cot^2(theta) = tan^2(theta) + cot^2(theta) + tan^2(theta) * cot^2(theta)

Step 4: Simplifying the equation further using trigonometric identities:

Using the identity, tan(theta) = sin(theta) / cos(theta), we can rewrite tan^2(theta) as:

tan^2(theta) = (sin^2(theta)) / (cos^2(theta))

Similarly, using the identity, cot(theta) = cos(theta) / sin(theta), we can rewrite cot^2(theta) as:

cot^2(theta) = (cos^2(theta)) / (sin^2(theta))

Substituting these expressions into the equation from Step 3:

(sin^2(theta) / cos^2(theta)) * (cos^2(theta) / sin^2(theta)) = (sin^2(theta) / cos^2(theta)) + (cos^2(theta) / sin^2(theta)) + (sin^2(theta) / cos^2(theta)) * (cos^2(theta) / sin^2(theta))

Simplifying the equation:

1 = (sin^2(theta)