Question

How do you solve \[\csc \theta - \sin \theta = \cos \theta \cot \theta \] ?

Answer 1

Hint:Here we need to prove that \[\csc \theta - \sin \theta = \cos \theta \cot \theta \], in which cosec function can be written in terms of sin and hence we can show that LHS = RHS.

Complete step by step answer:
The given equation is
\[\csc \theta - \sin \theta = \cos \theta \cot \theta \]
In which we need to prove that LHS terms are equal to RHS terms as given.
Let us consider the LHS terms of the given equation i.e.,
\[\csc \theta - \sin \theta \]
Here, \[\csc \theta \] can be written in terms of sin as
=\[\dfrac{1}{{\sin \theta }} - \sin \theta \]
Simplify the functions in terms of sin we get
\[ = \dfrac{{1 - {{\sin }^2}\theta }}{{\sin \theta }}\]
As we know that \[1 - {\sin ^2}\theta \]is \[{\cos ^2}\theta \], hence applying this in the equation becomes as
\[ = \dfrac{{{{\cos }^2}\theta }}{{\sin \theta }}\]
Hence, we get
\[ = \cos \theta \dfrac{{\cos \theta }}{{\sin \theta }}\]
As we know that \[\dfrac{{\cos \theta }}{{\sin \theta }}\] is \[\cot \theta \], hence applying this in the equation, we get
\[ = \cos \theta \cot \theta \]
Therefore, the term we got is equal to RHS i.e., \[\cos \theta \cot \theta \]
Hence, LHS = RHS
\[\csc \theta - \sin \theta = \cos \theta \cot \theta \]

Additional information:
In trigonometry sin, cos and tan values are the primary functions we consider while solving
trigonometric problems. These trigonometry values are used to measure the angles and sides of a right-angle triangle. Apart from sine, cosine and tangent values, other values are cotangent, secant and cosecant.

When we find sin, cos and tan values for a triangle, we usually consider these angles: 0°, 30°, 45°, 60° and 90°. The trigonometric values are about the knowledge of standard angles for a given triangle as per the trigonometric ratios. Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. These angles can also be represented in the form of radians such as 0, π/6, π/4, π/3, and π/2. These angles are most commonly and frequently used in trigonometry.

Note: The key point to find the values of any trigonometric function is to note the chart of all functions as shown and calculates all the terms asked. And here are some of the formulas to be noted.
\[\tan \theta \] is \[\dfrac{{\sin \theta }}{{\cos \theta }}\]
\[\cot \theta \] is \[\dfrac{{\cos \theta }}{{\sin \theta }}\]
\[\sin \theta \] is \[\dfrac{{\tan \theta }}{{\cot \theta }}\]
\[\cos \theta \] is \[\dfrac{{\sin \theta }}{{\tan \theta }}\]
\[\sec \theta \] is \[\dfrac{{\tan \theta }}{{\sin \theta }}\]
\[\csc \theta \] is \[\dfrac{{\sec \theta }}{{\tan \theta }}\]