Problem 56 - Precalculus Essentials

Question

Verify each identity. $$\left(\cot ^{2} \theta+1\right)\left(\sin ^{2} \theta+1\right)=\cot ^{2} \theta+2$$

Step-by-Step Solution

Answer

Upon substituting the expressions \(\cot^2 \theta\) and \(\sin^2 \theta\) in the given identity with relevant trigonometric identities and simplifying, it is verified both sides of the identity match, hence the identity is correct.

Step 1: Substitute the identities

Substitute \(\cot^2 \theta\) with \(\csc^2 \theta - 1\) and \(\sin^2 \theta\) with \(1 - \csc^2 \theta\) giving: \[(\csc^2 \theta - 1 + 1)(1 - \csc^2 \theta + 1)\]

Step 2: Simplify the expression

Simplify the above expression to: \[(\csc^2 \theta)(2 - \csc^2 \theta)\] By using the identity \(\csc^2 \theta = 1 + \cot^2 \theta\), replace all \(\csc^2 \theta\) terms and obtain: \[(1 + \cot^2 \theta)(2 - 1 - \cot^2 \theta)\]

Step 3: Simplify further

Further simplify to get: \[\cot^2 \theta + 2\] which is the right hand side of the given identity.

Step 4: Conclude

Both sides of the given identity now match, thus the identity is verified.