# Evaluate $\sin^4{x}$ $+$ $2\sin^2{x}\bigg(1-\dfrac{1}{\csc^2{x}}\bigg)$ $+$ $\cos^4{x}$

A trigonometric expression is defined in terms of three trigonometric functions sine, cosine and cosecant in terms of angle x in this trigonometric problem.

$\sin^4{x}$ $+$ $2\sin^2{x}\bigg(1-\dfrac{1}{\csc^2{x}}\bigg)$ $+$ $\cos^4{x}$

The sine raised to the power of $4$ of an angle $x$ is added to the product of two times sine squared of angle $x$ and the subtraction of the reciprocal of cosecant square of $x$ from $1$. The sum of them is added to the cosine raise to the power of $4$ of an angle $x$. the sum of the three trigonometric expressions has to evaluate by simplification in this trigonometric problem.

### Simplify the expression by multiplication

In the second term of the expression, two trigonometric functions are multiplied. The second factor in the second term is a difference of two terms. The difference of them is multiplied by a trigonometric function. The product of them can be evaluated by the distributive property of multiplication over subtraction.

$=\,\,\,$ $\sin^4{x}$ $+$ $2\sin^2{x} \times 1$ $-$ $2\sin^2{x} \times \dfrac{1}{\csc^2{x}}$ $+$ $\cos^4{x}$

### Simplify the expression by identities

The reciprocal of cosecant squared of angle $x$ can be converted in sine function as per the reciprocal identity of cosecant.

$=\,\,\,$ $\sin^4{x}$ $+$ $2\sin^2{x}$ $-$ $2\sin^2{x} \times \sin^2{x}$ $+$ $\cos^4{x}$

Now, concentrate on simplifying the trigonometric expression.

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

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

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

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

Express the cosine raised to the power of $4$ of an angle $x$ as the square of cosine square of angle $x$.

$=\,\,\,$ $\big(\cos^2{x}\big)^2$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

According to the cosine squared identity, the square of cosine can be written in terms of square of sine.

$=\,\,\,$ $\big(1-\sin^2{x}\big)^2$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

The expression in the first term represents the square of difference of two terms and it can be expanded as per square of difference formula.

$=\,\,\,$ $1^2$ $+$ $\big(\sin^2{x}\big)^2$ $-$ $2 \times 1 \times \sin^2{x}$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

### Evaluate the expression by simplification

Write all terms in a convenient order for simplifying the trigonometric expression and it is helpful for calculating the value of given trigonometric expression.

$=\,\,\,$ $1$ $+$ $\sin^4{x}$ $-$ $2\sin^2{x}$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

$=\,\,\,$ $1$ $+$ $\sin^4{x}$ $-$ $\sin^4{x}$ $-$ $2\sin^2{x}$ $+$ $2\sin^2{x}$

$=\,\,\,$ $1$ $+$ $\cancel{\sin^4{x}}$ $-$ $\cancel{\sin^4{x}}$ $-$ $\cancel{2\sin^2{x}}$ $+$ $\cancel{2\sin^2{x}}$

$=\,\,\,$ $1$

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