If cosec 39° = x, then the value of \(\frac 1 {\rm cosec^2\;51^\circ} + \sin^2 39^\circ + \tan^2 51^\circ - \frac 1 {\sin^2 51^\circ\sec^2 39^\circ}\) is:
x2 - 1
Evaluating Trigonometric Expression with Complementary Angles
The problem asks us to find the value of a given trigonometric expression, given that cosec 39° = x. The expression involves trigonometric ratios of angles 39° and 51°. Notice that these angles are complementary, as 39° + 51° = 90°. This suggests we should use complementary angle identities to simplify the expression.
Using Complementary Angle Identities
We can rewrite the trigonometric ratios of 51° in terms of 39° using the following identities:
- sin (90° - \(\theta\)) = cos \(\theta\)
- cos (90° - \(\theta\)) = sin \(\theta\)
- tan (90° - \(\theta\)) = cot \(\theta\)
- cot (90° - \(\theta\)) = tan \(\theta\)
- sec (90° - \(\theta\)) = cosec \(\theta\)
- cosec (90° - \(\theta\)) = sec \(\theta\)
Applying these identities for \(\theta\) = 39°, so that 90° - \(\theta\) = 51°:
- cosec 51° = cosec (90° - 39°) = sec 39°
- sin 51° = sin (90° - 39°) = cos 39°
- tan 51° = tan (90° - 39°) = cot 39°
Substituting into the Given Expression
The given expression is:
\(\frac 1 {\rm cosec^2\;51^\circ} + \sin^2 39^\circ + \tan^2 51^\circ - \frac 1 {\sin^2 51^\circ\sec^2 39^\circ}\)
Substitute the complementary angle forms:
- \(\frac 1 {\rm cosec^2\;51^\circ}\) = \(\frac 1 {\sec^2\;39^\circ}\) = cos² 39°
- sin² 39° remains as is.
- tan² 51° = cot² 39°
- \(\frac 1 {\sin^2 51^\circ\sec^2 39^\circ}\) = \(\frac 1 {\cos^2 39^\circ\sec^2 39^\circ}\). Since sec \(\theta\) = \(\frac{1}{\cos \theta}\), sec² 39° = \(\frac{1}{\cos^2 39^\circ}\). So, \(\cos^2 39^\circ \cdot \sec^2 39^\circ = \cos^2 39^\circ \cdot \frac{1}{\cos^2 39^\circ} = 1\). Therefore, \(\frac 1 {\cos^2 39^\circ\sec^2 39^\circ} = \frac 1 1 = 1\).
Now, substitute these simplified terms back into the expression:
The expression becomes: cos² 39° + sin² 39° + cot² 39° - 1
Applying Pythagorean Identities
Recall the Pythagorean identity: sin² \(\theta\) + cos² \(\theta\) = 1.
For \(\theta\) = 39°, we have sin² 39° + cos² 39° = 1.
Substitute this into the simplified expression:
(cos² 39° + sin² 39°) + cot² 39° - 1 = 1 + cot² 39° - 1
Simplifying further:
1 + cot² 39° - 1 = cot² 39°
Expressing the Result in Terms of x
We are given that cosec 39° = x. We need to find cot² 39° in terms of x.
Recall another Pythagorean identity: cosec² \(\theta\) - cot² \(\theta\) = 1.
Rearranging this identity to solve for cot² \(\theta\):
cot² \(\theta\) = cosec² \(\theta\) - 1
For \(\theta\) = 39°:
cot² 39° = cosec² 39° - 1
Since cosec 39° = x, cosec² 39° = x².
Therefore,
cot² 39° = x² - 1
The value of the given expression is x² - 1.
Step-by-Step Derivation
Let the given expression be E.
\(E = \frac 1 {\rm cosec^2\;51^\circ} + \sin^2 39^\circ + \tan^2 51^\circ - \frac 1 {\sin^2 51^\circ\sec^2 39^\circ}\)
Step 1: Use complementary angle identities to express terms in 51° using 39°.
- cosec 51° = sec 39°
- sin 51° = cos 39°
- tan 51° = cot 39°
Step 2: Substitute these into the expression.
\(E = \frac 1 {\sec^2\;39^\circ} + \sin^2 39^\circ + \cot^2 39^\circ - \frac 1 {\cos^2 39^\circ\sec^2 39^\circ}\)
Step 3: Use reciprocal identities \(\left(\sec \theta = \frac{1}{\cos \theta}\right)\).
\(E = \cos^2 39^\circ + \sin^2 39^\circ + \cot^2 39^\circ - \frac 1 {\cos^2 39^\circ \cdot \frac{1}{\cos^2 39^\circ}}\)
Step 4: Simplify the last term.
\(E = \cos^2 39^\circ + \sin^2 39^\circ + \cot^2 39^\circ - \frac 1 1\)
\(E = \cos^2 39^\circ + \sin^2 39^\circ + \cot^2 39^\circ - 1\)
Step 5: Use the Pythagorean identity sin² \(\theta\) + cos² \(\theta\) = 1.
\(E = (\cos^2 39^\circ + \sin^2 39^\circ) + \cot^2 39^\circ - 1\)
\(E = 1 + \cot^2 39^\circ - 1\)
Step 6: Simplify.
\(E = \cot^2 39^\circ\)
Step 7: Use the Pythagorean identity cosec² \(\theta\) - cot² \(\theta\) = 1 to relate cot² 39° to cosec² 39°.
cot² 39° = cosec² 39° - 1
Step 8: Substitute the given value cosec 39° = x.
cot² 39° = x² - 1
Thus, the value of the expression is x² - 1.
Revision Table: Trigonometric Identities Used
| Identity Type | Identity | Application in Problem |
|---|---|---|
| Complementary Angle | cosec (90° - \(\theta\)) = sec \(\theta\) | cosec 51° = sec 39° |
| Complementary Angle | sin (90° - \(\theta\)) = cos \(\theta\) | sin 51° = cos 39° |
| Complementary Angle | tan (90° - \(\theta\)) = cot \(\theta\) | tan 51° = cot 39° |
| Reciprocal | \(\sec \theta = \frac{1}{\cos \theta}\) | \(\frac{1}{\sec^2 39^\circ} = \cos^2 39^\circ\) |
| Reciprocal | \(\sec \theta = \frac{1}{\cos \theta}\) | \(\cos^2 39^\circ \sec^2 39^\circ = \cos^2 39^\circ \frac{1}{\cos^2 39^\circ} = 1\) |
| Pythagorean | sin² \(\theta\) + cos² \(\theta\) = 1 | sin² 39° + cos² 39° = 1 |
| Pythagorean | cosec² \(\theta\) - cot² \(\theta\) = 1 | cot² 39° = cosec² 39° - 1 |
Additional Information: Trigonometric Ratios of Complementary Angles
Two angles are called complementary if their sum is 90°. For any acute angle \(\theta\), the angle 90° - \(\theta\) is complementary to \(\theta\). The trigonometric ratios of complementary angles are related as follows:
- sin \((90^\circ - \theta) = \cos \theta\)
- cos \((90^\circ - \theta) = \sin \theta\)
- tan \((90^\circ - \theta) = \cot \theta\)
- cot \((90^\circ - \theta) = \tan \theta\)
- sec \((90^\circ - \theta) = \text{cosec } \theta\)
- cosec \((90^\circ - \theta) = \sec \theta\)
These identities are very useful in simplifying trigonometric expressions and solving trigonometric equations, especially when dealing with angles like (90° - \(\theta\)) or angles that sum up to 90°.
In this problem, the angles were 39° and 51°, which sum up to 90°, making them complementary. Recognizing this relationship was key to simplifying the expression using the above identities.
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