Question

How do you simplify \[\dfrac{{si{n^2}x - 1}}{{1 + {{\sin }^2}x}}?\]

Answer 1

Hint: In this question we have simplified the given trigonometric form. Next, we use some trigonometric identities and then simplify to arrive at our final answer. Next, we rearrange the trigonometric functions .And also we are going to division and LCM in complete step by step solution.
The trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

Complete step by step answer:
Given,
\[ \Rightarrow \dfrac{{si{n^2}x - 1}}{{1 + {{\sin }^2}x}}\]
First, use the Pythagorean trigonometric identity,\[{\sin ^2}(x) + {\cos ^2}(x) = 1\], to simplify "\[si{n^2}(x)\]" in the numerator.
Rearrange the Pythagorean identity \[{\sin ^2}(x) + {\cos ^2}(x) = 1\] to isolate cos2x:
\[ \Rightarrow {\cos ^2}(x) = 1 - {\sin ^2}(x)\]
Hence, \[{\cos ^2}(x) = 1 - {\sin ^2}(x)\]
Now, substitute the given trigonometry form and we get,
\[ \Rightarrow \dfrac{{(1 - co{s^2}x) - 1}}{{1 + si{n^2}x}}\]
Next, cancelling number 1 from the above term and we get the final answer
\[ \Rightarrow \dfrac{{{1} - co{s^2}x - {1}}}{{1 + si{n^2}x}}\]
\[ \Rightarrow \dfrac{{ - co{s^2}x}}{{1 + si{n^2}x}}\]
This is the required answer for the given trigonometry term.

Note: we remember that the trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.
Sine Function : \[\sin \left( \theta \right) = \dfrac{{Opposite}}{{Hypotenuse}}\]
Cosine Function : \[\cos \left( \theta \right) = \dfrac{{Adjacent}}{{Hypotenuse}}\]
Tangent Function: \[\tan \left( \theta \right) = \dfrac{{Opposite}}{{Adjacent}}\]
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, \[{90^ \circ }\] or $\dfrac{\pi }{2}$radians.
Trigonometry simply means calculations with triangles (that’s where the tri comes from). It is a study of relationships in mathematics involving lengths, heights and angles of different triangles. The field emerged during the 3rd century BC, from applications of geometry to astronomical studies. Trigonometry spreads its applications into various fields such as architects, surveyors, astronauts, physicists, engineers and even crime scene investigators.