Understanding the Trigonometric Expression

We are asked to find the value of a given trigonometric expression. This expression involves various trigonometric ratios and angles. To solve this, we will simplify the numerator and the denominator separately using trigonometric identities and properties of complementary angles.

The expression is:

\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)

Simplifying the Numerator Part of the Expression

Let's first look at the numerator:

\(\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ\)

We can rearrange the terms to group them effectively:

\((\sin 23^\circ \cos 67^\circ + \cos 23^\circ \sin 67^\circ) + (\sec52^\circ \sin38^\circ + \rm cosec52^\circ \cos 38^\circ)\)

Part 1: Simplifying the First Group in Numerator

The first group is \(\sin 23^\circ \cos 67^\circ + \cos 23^\circ \sin 67^\circ\). This matches the sine addition formula:

\(\sin(A+B) = \sin A \cos B + \cos A \sin B\)

Here, \(A = 23^\circ\) and \(B = 67^\circ\).

So, \(\sin 23^\circ \cos 67^\circ + \cos 23^\circ \sin 67^\circ = \sin(23^\circ + 67^\circ) = \sin(90^\circ)\).

The value of \(\sin(90^\circ)\) is 1.

Thus, the first group simplifies to 1.

Part 2: Simplifying the Second Group in Numerator

The second group is \(\sec52^\circ \sin38^\circ + \rm cosec52^\circ \cos 38^\circ\).

We can use the concept of complementary angles. Remember that if \(A + B = 90^\circ\), then \(\sin A = \cos B\), \(\cos A = \sin B\), \(\tan A = \cot B\), \(\cot A = \tan B\), \(\sec A = \rm cosec B\), and \(\rm cosec A = \sec B\).

Here, \(52^\circ + 38^\circ = 90^\circ\).

• \(\sin 38^\circ = \sin(90^\circ - 52^\circ) = \cos 52^\circ\)
• \(\cos 38^\circ = \cos(90^\circ - 52^\circ) = \sin 52^\circ\)
• \(\sec 52^\circ = \frac{1}{\cos 52^\circ}\)
• \(\rm cosec 52^\circ = \frac{1}{\sin 52^\circ}\)

Substitute these into the second group expression:

\(\sec52^\circ \sin38^\circ + \rm cosec52^\circ \cos 38^\circ = \left(\frac{1}{\cos 52^\circ}\right) (\cos 52^\circ) + \left(\frac{1}{\sin 52^\circ}\right) (\sin 52^\circ)\)

\(= 1 + 1 = 2\)

Thus, the second group simplifies to 2.

Total Numerator Value Calculation

The total value of the numerator is the sum of the simplified values from Part 1 and Part 2.

Numerator = (Value of Part 1) + (Value of Part 2) = \(1 + 2 = 3\).

So, the numerator is 3.

Simplifying the Denominator Part of the Expression

Now let's look at the denominator:

\(\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ\)

We can again use the concept of complementary angles. Note that \(20^\circ + 70^\circ = 90^\circ\).

So, \(\tan 70^\circ = \tan(90^\circ - 20^\circ) = \cot 20^\circ\).

Substitute this into the denominator expression:

\(\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ = \rm cose{c^2}20^\circ - \cot^2 20^\circ\)

This expression matches a fundamental trigonometric identity:

\(\rm cosec^2 \theta - \cot^2 \theta = 1\)

Here, \(\theta = 20^\circ\).

So, \(\rm cose{c^2}20^\circ - \cot^2 20^\circ = 1\).

Thus, the denominator simplifies to 1.

Calculating the Final Expression Value

The value of the original expression is the numerator divided by the denominator.

Expression Value = \(\frac{\text{Numerator}}{\text{Denominator}} = \frac{3}{1} = 3\)

Therefore, the value of the given trigonometric expression is 3.

Revision Table: Key Trigonometric Concepts Review

Additional Information on Trigonometric Identities and Angles

Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. They are crucial tools for simplifying complex trigonometric expressions and solving trigonometric equations.

We used the complementary angle relationships extensively in this problem. These relationships arise directly from the definition of trigonometric ratios in a right-angled triangle. If one acute angle is \(\theta\), the other acute angle is \(90^\circ - \theta\).

Let's list some key complementary angle relations used in trigonometry:

• \(\sin(90^\circ - \theta) = \cos \theta\)
• \(\cos(90^\circ - \theta) = \sin \theta\)
• \(\tan(90^\circ - \theta) = \cot \theta\)
• \(\cot(90^\circ - \theta) = \tan \theta\)
• \(\sec(90^\circ - \theta) = \rm cosec \theta\)
• \(\rm cosec(90^\circ - \theta) = \sec \theta\)

The Pythagorean identities are also fundamental. The three main Pythagorean identities are:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\) (or \(\sec^2 \theta - \tan^2 \theta = 1\))
• \(1 + \cot^2 \theta = \rm cosec^2 \theta\) (or \(\rm cosec^2 \theta - \cot^2 \theta = 1\))

Mastering these identities and relationships is key to successfully tackling trigonometry problems like the one solved here.