Question

Simplifying Trigonometric Expressions-

Simplify the trigonometric expression$\left(si{n}^4\alpha -co{s}^4\alpha +co{s}^2\alpha \right)$

Short answer

On simplifying expression we get$\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\alpha }$

Step-by-step solution

Step 1. Given information

A trigonometric expression as$\left({\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha \right)$

Step 2. Concept used

To simplify the expression above, use trigonometric identities. After the simplification, the outcome will be the answer.

Step 3. Calculation

Now, simplify $\left({\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha \right)$using identities:

Convert ${\sin }^4\alpha ={\left(1-{\cos }^2\alpha \right)}^2$

$\begin{array}{l}{\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha ={\left(1-{\cos }^2\alpha \right)}^2-{\cos }^4\alpha +{\cos }^2\alpha \\ {\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha =1+{\cos }^4\alpha -2{\cos }^2\alpha -{\cos }^4\alpha +{\cos }^2\alpha \\ {\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha =1-{\cos }^2\alpha \end{array}$

Now use formula $\left(1-{\cos }^2\alpha \right)={\sin }^2\alpha$

${\sin }^4\alpha -{\cos }^4\alpha +{\cos }^2\alpha ={\sin }^2\alpha$