Question

Proving more Identities

Verify the identity$\mathbf{(}\mathbf{sin\alpha }\mathbf{-}\mathbf{tan\alpha }\mathbf{)}\mathbf{(}\mathbf{cos\alpha }\mathbf{-}\mathbf{cot\alpha }\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{cos\alpha }\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{sin\alpha }\mathbf{-}\mathbf{1}\mathbf{)}$

Short answer

The expression$\mathbf{(}\mathbf{sin\alpha }\mathbf{-}\mathbf{tan\alpha }\mathbf{)}\mathbf{(}\mathbf{cos\alpha }\mathbf{-}\mathbf{cot\alpha }\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{cos\alpha }\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{sin\alpha }\mathbf{-}\mathbf{1}\mathbf{)}$ is an identity.

Step-by-step solution

Step 1. Given information

An expression$(\sin \alpha -\tan \alpha )(\cos \alpha -\cot \alpha )=(\cos \alpha -1)(\sin \alpha -1)$

Step 2. Concept used

Divide the sentence into two halves, LHS and RHS, to evaluate if it has a distinct identity. After that, make both of them as simple as possible. If the results are the same, the phrase is deemed an identity.

Step 3. Calculation

Now, simplify LHS of $(\sin \alpha -\tan \alpha )(\cos \alpha -\cot \alpha )=(\cos \alpha -1)(\sin \alpha -1)$

$\begin{array}{l}(\sin \alpha -\tan \alpha )(\cos \alpha -\cot \alpha )=\left(\sin \alpha -\frac{\sin \alpha }{\cos \alpha }\right)\left(\cos \alpha -\frac{\cos \alpha }{\sin \alpha }\right)\\ \because \tan \alpha =\frac{\sin \alpha }{\cos \alpha }\&\cot \alpha =\frac{\cos \alpha }{\sin \alpha }\\ =\sin \alpha \cos \alpha -\sin \alpha -\cos \alpha +1\end{array}$

Now factorize and take common,

$\begin{array}{l}=\sin \alpha (\cos \alpha -1)-(\cos \alpha -1)\\ =(\sin \alpha -1)(\cos \alpha -1)\end{array}$

RHS of the equation is$(\sin \alpha -1)(\cos \alpha -1)$

Both RHS and LHS are equal. Hence it is an identity