Thrice sine of angle seventy two levels divided by cosine of angle eighteen levels minus secant of angle thirty two levels divided by cosecant of angle fifty eight levels is a given trigonometric expression on this trigonometry downside.

The angles in sine, cosine, secant and cosecant features are usually not normal angles. So, it’s onerous to recollect the values of sine of angle $72^circ$, cosine of angle $18^circ$, secant of angle $32^circ$ and cosecant of angle $58^circ$. Nonetheless, the trigonometric expression will be evaluated with out substituting their values.

Calculate the worth of the primary time period

The angles contained in the sine and cosine features within the first time period are seventy two and eighteen levels respectively. The sum of them is a proper angle. In different phrases, $72^circ +18^circ ,=, 90^circ$. Due to this fact, the angles of the trigonometric features within the first time period are complementary angles.

$=,,,$ $3 instances dfrac{sin{72^circ}}{cos{18^circ}}$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

The angle eighteen levels will be written because the subtraction of angle seventy two levels from the angle ninety levels.

$=,,,$ $3 instances dfrac{sin{72^circ}}{cos{(90^circ-72^circ)}}$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

In keeping with the cofunction id of cosine perform, the cosine of ninety levels minus seventy two is the same as the sine of angle seventy two levels.

$=,,,$ $3 instances dfrac{sin{72^circ}}{sin{(72^circ)}}$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

It’s time to simplify the trigonometric expression within the first time period to seek out its worth.

$=,,,$ $3 instances dfrac{sin{72^circ}}{sin{72^circ}}$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

$=,,,$ $3 instances dfrac{cancel{sin{72^circ}}}{cancel{sin{72^circ}}}$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

$=,,,$ $3 instances 1$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

$=,,,$ $3$ $-$ $dfrac{sec{32^circ}}{csc{58^circ}}$

Consider the worth of the second time period

Now, let’s deal with discovering the trigonometric expression within the second time period. In actual fact, the trigonometric identities in secant and cosecant features are usually not all the time helpful when their use is in contrast with the usage of their reciprocal trigonometric features. Due to this fact, it’s all the time higher to specific secant and cosecant features of their reciprocal type.

$=,,,$ $3$ $-$ $dfrac{sec{32^circ} instances 1}{csc{58^circ}}$

Now, break up the second time period as a product of two features as per the multiplication of the fractions.

$=,,,$ $3$ $-$ $sec{32^circ}$ $instances$ $dfrac{1}{csc{58^circ}}$

In keeping with the reciprocal id of cosecant perform, the cosecant of angle fifty eight levels is written as sine of angle fifty eight levels.

$=,,,$ $3$ $-$ $sec{32^circ}$ $instances$ $sin{58^circ}$

In keeping with the reciprocal id of cosine perform, the secant of angle thirty two levels will be expressed as a reciprocal of sine of angle thirty two levels.

$=,,,$ $3$ $-$ $dfrac{1}{cos{32^circ}}$ $instances$ $sin{58^circ}$

Now, multiply each features as per the multiplication of the fractions.

$=,,,$ $3$ $-$ $dfrac{1 instances sin{58^circ}}{cos{32^circ}}$

$=,,,$ $3$ $-$ $dfrac{sin{58^circ}}{cos{32^circ}}$

The angles insides the sine and cosine features are fifty eight and thirty two levels respectively, and their sum is ninety levels. It means, $58^circ+32^circ ,=, 90^circ$. So, the angle fifty eight levels will be written because the subtraction of thirty two levels from ninety levels.

$=,,,$ $3$ $-$ $dfrac{sin{(90^circ-32^circ)}}{cos{32^circ}}$

In keeping with the cofunction id of sine perform, the sine of angle ninety levels minus thirty two levels is the same as the cosine of angle thirty two levels.

$=,,,$ $3$ $-$ $dfrac{cos{32^circ}}{cos{(32^circ)}}$

$=,,,$ $3$ $-$ $dfrac{cos{32^circ}}{cos{32^circ}}$

$=,,,$ $3$ $-$ $dfrac{cancel{cos{32^circ}}}{cancel{cos{32^circ}}}$

Discover the worth of expression by simplification

The trigonometric expression is efficiently simplified as an arithmetic expression and it’s time to discover its worth mathematically.

$=,,,$ $3-1$

$=,,,$ $2$

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