Simplify Trigonometric Expression Step-by-Step
The problem requires simplifying the given trigonometric expression:
$ \frac{(4 + 4 \cot^2 A) \tan A}{(3 + 3 \tan^2 A) \operatorname{cosec} A} $Factor and Apply Identities
First, factor out the common constants from the numerator and the denominator:
$ \frac{4(1 + \cot^2 A) \tan A}{3(1 + \tan^2 A) \operatorname{cosec} A} $Now, apply the Pythagorean trigonometric identities:
- $1 + \cot^2 A = \operatorname{cosec}^2 A$
- $1 + \tan^2 A = \sec^2 A$
Substitute these identities into the expression:
$ \frac{4(\operatorname{cosec}^2 A) \tan A}{3(\sec^2 A) \operatorname{cosec} A} $Reduce the Expression
Cancel out one $\operatorname{cosec} A$ term from the numerator and denominator:
$ \frac{4 \operatorname{cosec} A \tan A}{3 \sec^2 A} $Convert to Sine and Cosine
Rewrite the expression in terms of $\sin A$ and $\cos A$ using the reciprocal and ratio identities:
- $\operatorname{cosec} A = \frac{1}{\sin A}$
- $\tan A = \frac{\sin A}{\cos A}$
- $\sec A = \frac{1}{\cos A} \implies \sec^2 A = \frac{1}{\cos^2 A}$
Substitute these into the expression:
$ \frac{4 \left(\frac{1}{\sin A}\right) \left(\frac{\sin A}{\cos A}\right)}{3 \left(\frac{1}{\cos^2 A}\right)} $Final Simplification
Simplify the numerator and the denominator separately:
- Numerator: $4 \left(\frac{1}{\sin A}\right) \left(\frac{\sin A}{\cos A}\right) = 4 \left(\frac{1}{\cos A}\right) = \frac{4}{\cos A}$
- Denominator: $3 \left(\frac{1}{\cos^2 A}\right) = \frac{3}{\cos^2 A}$
Now divide the simplified numerator by the simplified denominator:
$ \frac{\frac{4}{\cos A}}{\frac{3}{\cos^2 A}} $To divide, multiply the numerator by the reciprocal of the denominator:
$ \frac{4}{\cos A} \times \frac{\cos^2 A}{3} $Simplify further:
$ \frac{4 \cos^2 A}{3 \cos A} = \frac{4 \cos A}{3} $The simplified expression is $\frac{4 \cos A}{3}$.
Important Questions from Trigonometric Identities
What is cos 2β equal to ?
What is the value of sec2γ?
On simplifying \(\frac{{{{\sin }^3}{\rm{A}} + \sin 3{\rm{\;A}}}}{{\sin {\rm{A}}}} + \frac{{{{\cos }^3}{\rm{A}} - \cos 3{\rm{\;A}}}}{{\cos {\rm{A}}}}\) we get
(1 – sin A + cos A) 2is equal to
What is \(\frac{{\cos {\rm{\theta }}}}{{1 - \tan {\rm{\theta }}}} + \frac{{\sin {\rm{\theta }}}}{{1 - \cot {\rm{\theta }}}}\) equal to?
