Question

How do you prove this trigonometric expression show your work on paper

Answer 1

Solution!!

$\sf \cos ^{2}\theta =\dfrac{\csc \theta \times \cos \theta }{\tan \theta + \cot \theta }$

Taking RHS

$\sf =\dfrac{\csc \theta \times \cos \theta }{\tan \theta + \cot \theta }$

Use the following relations to transform the expressions:

→ $\sf \csc \theta = \dfrac{1}{\sin \theta }$

→ $\sf \tan \theta = \dfrac{\sin \theta }{\cos \theta }$

→ $\sf \cot \theta = \dfrac{\cos \theta }{\sin \theta }$

$\sf =\dfrac{\frac{1}{\sin \theta }\times \cos \theta }{\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }}$

Calculate the product.

$\sf =\dfrac{\frac{\cos \theta }{\sin \theta }}{\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }}$

Write all the numerators above the common denominator.

$\sf =\dfrac{\frac{\cos \theta }{\sin \theta }}{\frac{\sin ^{2}\theta + \cos ^{2}\theta }{\cos \theta \sin \theta }}$

Use the following to simplify the expression:

→ $\sf \sin ^{2}\theta + \cos ^{2}\theta =1$

$\sf =\dfrac{ \frac{\cos \theta }{\sin \theta }}{\frac{1}{\cos \theta \sin \theta }}$

Simplify the complex fraction.

$\sf =\cos \theta \times \cos \theta$

$\sf =\cos ^{2}\theta$

LHS = RHS

Hence, proved.