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Chapter 5: Problem 56

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

Short Answer

Expert verified
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 by step solution

01

Substitute the identities

Substitute \(\cot^2 \theta\) with \(\csc^2 \theta - 1\) and \(\sin^2 \theta\) with \(1 - \csc^2 \theta\) giving: \[(\csc^2 \theta - 1 + 1)(1 - \csc^2 \theta + 1)\]
02

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: \[(1 + \cot^2 \theta)(2 - 1 - \cot^2 \theta)\]
03

Simplify further

Further simplify to get: \[\cot^2 \theta + 2\] which is the right hand side of the given identity.
04

Conclude

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

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