Question

Solve the question quickly, please!
Using Trigonometric identities.

Answer 1

Let : I'm taking x in place of theta

To prove :  { ( 1+cot²x) tan x } ÷ sec²x = cot x

Identity used :

• cosec²∅ = 1 + cot²∅

• tan ∅ = sin ∅ ÷ cos ∅

• cosec ∅ = 1 ÷ sin ∅

• cot ∅ = cos ∅ ÷ sin ∅

• sec ∅ = 1 ÷ cos ∅

• $\dfrac{\frac{a}{b} }{\frac{c}{d} }\: = \:\dfrac{a\times d}{b\times c}$

Solution :

LHS

$\dfrac{(1 + { \cot(x) }^{2}) \tan(x) }{ { \sec(x) }^{2} } \\ \\ \\ = \dfrac{ { \csc(x)^{2} \times \tan(x) } }{ { \sec(x) }^{2} } \\ \\\\ = \dfrac{ \dfrac{1}{ { \sin(x) }^{2} } \times \dfrac{ \sin(x) }{ \cos(x) } }{ \dfrac{1}{ { \cos(x) }^{2} } } \\\\ \\ = \dfrac{ { \cos(x) }^{2} }{ \sin(x) \cos(x) } \\\\ \\ = \dfrac{ \cos(x) }{ \sin(x) } \\ \\ = \cot(x)$

RHS = cot x

LHS = RHS

HENCE PROVED