Simplify: \( sec^{2} \phi cot^{2 } \phi - \frac{1}{ tan^{2} \phi } +2( \frac{1}{ cosec^{2} \phi }+ \frac{1}{ sec^{2} \phi } )\)
Simplifying the Trigonometric Expression
The question asks us to simplify the following trigonometric expression:
$ sec^{2} \phi \cot^{2 } \phi - \frac{1}{ tan^{2} \phi } +2 \left( \frac{1}{ cosec^{2} \phi }+ \frac{1}{ sec^{2} \phi } \right) $
Key Trigonometric Identities
To simplify this expression, we will use the following fundamental trigonometric identities:
- \( \sec \phi = \frac{1}{\cos \phi} \)
- \( \csc \phi = \frac{1}{\sin \phi} \)
- \( \tan \phi = \frac{\sin \phi}{\cos \phi} \)
- \( \cot \phi = \frac{\cos \phi}{\sin \phi} \)
- \( \cot \phi = \frac{1}{\tan \phi} \)
- \( \sin^{2} \phi + \cos^{2} \phi = 1 \)
- \( \csc^{2} \phi - \cot^{2} \phi = 1 \)
Step-by-Step Simplification
Step 1: Simplify the first term, \( \sec^{2} \phi \cot^{2 } \phi \)
We can rewrite this term using the definitions of secant and cotangent:
$ \sec^{2} \phi \cot^{2 } \phi = \left( \frac{1}{\cos^{2} \phi} \right) \left( \frac{\cos^{2} \phi}{\sin^{2} \phi} \right) $Cancel out \( \cos^{2} \phi \):
$ = \frac{1}{\sin^{2} \phi} $Using the definition of cosecant, this simplifies to:
$ = \csc^{2} \phi $Step 2: Simplify the second term, \( \frac{1}{ \tan^{2} \phi } \)
Using the identity \( \cot \phi = \frac{1}{\tan \phi} \), this term becomes:
$ \frac{1}{ \tan^{2} \phi } = \cot^{2} \phi $Step 3: Simplify the third term, \( 2 \left( \frac{1}{ \cosec^{2} \phi }+ \frac{1}{ \sec^{2} \phi } \right) \)
Rewrite \( \frac{1}{\csc^{2} \phi} \) and \( \frac{1}{\sec^{2} \phi} \) using the definitions of sine and cosine:
$ \frac{1}{ \cosec^{2} \phi } = \sin^{2} \phi $ $ \frac{1}{ \sec^{2} \phi } = \cos^{2} \phi $Substitute these back into the third term:
$ 2 \left( \sin^{2} \phi + \cos^{2} \phi \right) $Now, apply the fundamental Pythagorean identity \( \sin^{2} \phi + \cos^{2} \phi = 1 \):
$ 2 (1) = 2 $Step 4: Combine the simplified terms
Substitute the simplified terms back into the original expression:
$ \underbrace{\csc^{2} \phi}_{\text{Term 1}} - \underbrace{\cot^{2} \phi}_{\text{Term 2}} + \underbrace{2}_{\text{Term 3}} $The expression is now:
$ \csc^{2} \phi - \cot^{2} \phi + 2 $Step 5: Apply the final identity
Use the identity \( \csc^{2} \phi - \cot^{2} \phi = 1 \):
$ (1) + 2 $Calculate the final result:
$ = 3 $Conclusion
The simplified value of the given trigonometric expression is 3.
Important Questions from Trigonometric Identities
- If $\sec\theta + \tan\theta = s$, then find the value of $\frac{s^2-1}{s^2+1}$.
- Find the value of $\frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} + \sqrt{\text{cosec}^2 A - 1}$.
- Solve the following:
$(\text{cosec}23^{\circ}-\text{sin}23^{\circ}) (\text{tan}23^{\circ} + \text{cot}23^{\circ}) (\text{sec}23^{\circ}-\text{cos}23^{\circ})$ If $M = \sqrt{\frac{1+\cos A}{1-\cos A}} - \text{cosec} A - \cot A$, and A is an acute angle, then value of M is
- If
$x = a \text{ cosec}A \text{ sin} B,$
$y = b \text{ cosec}A \text{ cos} B,$ and
$z = c \text{ cot} A,$
then, what is the value of $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}$?
