Question

Prove that \[\cot \theta - \cot 2\theta = \cos ec2\theta \]

Answer 1

Hint: Use the various general trigonometric formulas as \[\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}\], and also the formula of \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] and \[\sin 2x = 2\sin x\cos x\]. We just need to expand the given term and if possible convert it to the similar denominator so that the numerator can be easily calculated.

Complete step-by-step answer:
As given that\[\cot \theta - \cot 2\theta \],
\[\cot \theta - \cot 2\theta \]
On using \[\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}\], we get,
\[ = \dfrac{{\cos \theta }}{{\sin \theta }} - \dfrac{{\cos 2\theta }}{{\sin 2\theta }}\]
On further simplification, use \[\sin 2x = 2\sin x\cos x\]and \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] putting it in the above equation, we get,
\[ = \dfrac{{\cos \theta }}{{\sin \theta }} - \dfrac{{{{\cos }^2}\theta - {{\sin }^2}x}}{{2\sin \theta \cos \theta }}\]
Now multiply the denominator and numerator of first term with \[2\cos \theta \] in order to make them similar,
\[ = \dfrac{{2\cos \theta \cos \theta }}{{2\cos \theta \sin \theta }} - \dfrac{{{{\cos }^2}\theta - {{\sin }^2}x}}{{2\sin \theta \cos \theta }}\]
On further simplification we get,
\[
   = \dfrac{{2{{\cos }^2}\theta - ({{\cos }^2}\theta - {{\sin }^2}\theta )}}{{2\cos \theta \sin \theta }} \\
   = \dfrac{{{{\cos }^2}\theta + {{\sin }^2}\theta }}{{2\cos \theta \sin \theta }} \\
  \]
Using the identity\[{\cos ^2}\theta + {\sin ^2}\theta = 1\]
\[ = \dfrac{1}{{2\cos \theta \sin \theta }}\]
Now as \[\sin 2x = 2\sin x\cos x\], we get,
\[ = \dfrac{1}{{\sin 2\theta }}\]
As, \[\cos ec\theta = \dfrac{1}{{\sin \theta }}\],
\[ = \cos ec2\theta \]
Hence, \[\cot \theta - \cot 2\theta = \cos ec2\theta \]
Hence, proved.

Additional information:
Manufacturing Industry. Trigonometry plays a major role in industry, where it allows manufacturers to create everything from automobiles to zigzag scissors. Engineers rely on trigonometric relationships to determine the sizes and angles of mechanical parts used in machinery, tools and equipment.

Note: There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are used in a short form as sin, cos, tan, csc, sec, cot. These are referred to as ratios since they can be expressed in terms of the sides of a right-angled triangle for a specific angle \[\theta \].