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Evaluate $\dfrac{\sin{3x}}{\sin{x}}$ $-$ $\dfrac{\cos{3x}}{\cos{x}}$ without using triple angle identities

A trigonometric expression is formed in this trigonometry problem by the involvement of sine triple and cosine triple angle functions. It can be evaluated by the triple angle identities and you have already learned how to find it.

Now, let us learn how to find the value of sine of three times $x$ divided by sine of angle $x$ minus cosine of three times $x$ divided by cosine of angle $x$ without using the triple angle trigonometric identities in this method.

Find the difference of Trigonometric fractions

The given trigonometric expression represents the difference of two fractions. So, let’s perform the subtraction of the fractions to find the difference of them.

$\implies$ $\dfrac{\sin{3x}}{\sin{x}}$ $-$ $\dfrac{\cos{3x}}{\cos{x}}$ $\,=\,$ $\dfrac{\sin{3x} \times \cos{x}-\cos{3x} \times \sin{x}}{\sin{x} \times \cos{x}}$

$=\,\,$ $\dfrac{\sin{3x}\cos{x}-\cos{3x}\sin{x}}{\sin{x}\cos{x}}$

Simplify the expression by an identity

The trigonometry expression in the numerator of the rational expression expresses the expansion of sine of difference of the angles. According to the sine angles difference identity, the trigonometric expression can be simplified.

$\sin{(3x-x)}$ $\,=\,$ $\sin{3x}\cos{x}$ $-$ $\cos{3x}\sin{x}$

Therefore, the trigonometric expression in the numerator can be simplified by its equivalent value.

$\implies$ $\dfrac{\sin{3x}\cos{x}-\cos{3x}\sin{x}}{\sin{x}\cos{x}}$ $\,=\,$ $\dfrac{\sin{(3x-x)}}{\sin{x}\cos{x}}$

Now, let us focus on simplifying the trigonometric expression in rational form further.

$=\,\,$ $\dfrac{\sin{(2x)}}{\sin{x}\cos{x}}$

$=\,\,$ $\dfrac{\sin{2x}}{\sin{x}\cos{x}}$

Find the value of the Trigonometric expression

The trigonometric expression in rational form can be further simplified by expanding the sine of double angle function $\sin{2x}$ as per the sine double angle identity.

$=\,\,$ $\dfrac{2\sin{x}\cos{x}}{\sin{x}\cos{x}}$

$=\,\,$ $\dfrac{2\cancel{\sin{x}\cos{x}}}{\cancel{\sin{x}\cos{x}}}$

$=\,\,$ $2$

Another method

Find $\dfrac{\sin{3x}}{\sin{x}}$ $-$ $\dfrac{\cos{3x}}{\cos{x}}$

Learn how to evaluate the value of sine of three times angle $x$ divided by sine of $x$ minus cosine of three times $x$ divided by cosine of angle $x$ by using the triple angle trigonometric identities.

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