Question

Proving Identities

Verify the identity${\cot }^{2}t-{\cos }^2={\cot }^{2}t{\cos }^{2}t$

Short answer

The expression${\cot }^{2}t-{\cos }^2={\cot }^{2}t{\cos }^{2}t$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression${\cot }^{2}t-{\cos }^2={\cot }^{2}t{\cos }^{2}t$

Step 2. Concept used.

Check it by dividing the phrase into two parts: LHS and RHS. Then, on their own, simplify both words. The phrase is an identity if both responses are equal.

Step 3. Calculation.

Now, simplify LHS of ${\cot }^{2}t-{\cos }^2={\cot }^{2}t{\cos }^{2}t$:

$$\begin{gathered} \mathrm{Put}{\cot }^{2}t=\frac{{\cos }^{2}t}{{\sin }^{2}t} \\ {\cot }^{2}t-{\cos }^{2}t=\frac{{\cos }^{2}t}{{\sin }^{2}t}-{\cos }^{2}t \\ =\frac{{\cos }^{2}t-{\cos }^{2}t{\sin }^{2}t}{{\sin }^{2}t} \\ =\frac{{\cos }^{2}t{\cos }^{2}t}{{\sin }^{2}t} \\ =\frac{{\cos }^{2}t}{{\sin }^{2}t}{\cos }^{2}t \\ ={\cot }^{2}t{\cos }^{2}t \end{gathered}$$

RHS of the equation is${\cot }^{2}t{\cos }^{2}t$

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