The given expression [tex]\dfrac{\frac{cos \theta}{sin \theta} + \frac{1}{sin\theta} }{\frac{1}{sin \theta} (1+cos \theta)} =1[/tex] is true
Proving trigonometry identity
Given the expression
[tex]\frac{cot \theta + cosec \theta}{cosec \theta(1+\frac{cot \theta}{cosec \theta} )} =1[/tex]
Note that:
- cotθ = cosθ/sinθ
- cosecθ = 1/sinθ
Substituting the given parameters into the formula;
[tex]\frac{\frac{cos \theta}{sin \theta} + \frac{1}{sin\theta} }{\frac{1}{sin \theta} (1+cos \theta)} =1[/tex]
Find the LCM of both the numerator and denominator
[tex]= \dfrac{\frac{cos \theta + 1}{sin \theta} }{(\frac{1+cos \theta}{sin \theta} )}\\[/tex]
Divide the result to have:
[tex]= \frac{1+cos \theta}{sin \theta} \times \frac{sin \theta}{1+cos \theta}\\ = \frac{1}{1}\\ = 1 (Proved)[/tex]
Learn more on proofs of trigonometry identity here: https://brainly.com/question/7331447
