Solving Trigonometric Expression: \(\frac{1}{{\sin 10^\circ }} - \frac{{\sqrt 3 }}{{\cos 10^\circ }}\)

Let's simplify the given trigonometric expression: \(\frac{1}{{\sin 10^\circ }} - \frac{{\sqrt 3 }}{{\cos 10^\circ }}\). To combine these terms, we first find a common denominator.

The common denominator for \(\sin 10^\circ\) and \(\cos 10^\circ\) is \(\sin 10^\circ \cos 10^\circ\). So, we rewrite the expression:

\(\frac{1}{{\sin 10^\circ }} - \frac{{\sqrt 3 }}{{\cos 10^\circ }} = \frac{1 \cdot \cos 10^\circ}{{\sin 10^\circ \cos 10^\circ}} - \frac{{\sqrt 3} \cdot \sin 10^\circ}{{\cos 10^\circ \sin 10^\circ}}\)

Combine the fractions:

\(= \frac{{\cos 10^\circ - \sqrt 3 \sin 10^\circ}}{{\sin 10^\circ \cos 10^\circ}}\)

Now, let's look at the numerator, \(\cos 10^\circ - \sqrt 3 \sin 10^\circ\). This form resembles the expansion of \(\cos(A+B)\) or \(\sin(A-B)\). We can use a standard technique involving multiplying and dividing by a constant to transform the numerator into a single trigonometric function.

Consider the expression \(a \cos \theta + b \sin \theta\). We can write it as \(R \cos(\theta - \alpha)\) or \(R \sin(\theta + \alpha)\), where \(R = \sqrt{a^2 + b^2}\). In our numerator, the coefficients are \(a = 1\) (for \(\cos 10^\circ\)) and \(b = -\sqrt 3\) (for \(\sin 10^\circ\)).

Calculate \(R\):

\(R = \sqrt{1^2 + (-\sqrt 3)^2} = \sqrt{1 + 3} = \sqrt 4 = 2\)

Now, factor out R=2 from the numerator:

\(\cos 10^\circ - \sqrt 3 \sin 10^\circ = 2 \left( \frac{1}{2} \cos 10^\circ - \frac{\sqrt 3}{2} \sin 10^\circ \right)\)

We know that \(\frac{1}{2} = \cos 60^\circ\) and \(\frac{\sqrt 3}{2} = \sin 60^\circ\). Substitute these values:

\(= 2 (\cos 60^\circ \cos 10^\circ - \sin 60^\circ \sin 10^\circ)\)

This expression inside the parentheses matches the cosine addition formula: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). Here, \(A = 60^\circ\) and \(B = 10^\circ\).

So the numerator becomes:

\(2 \cos (60^\circ + 10^\circ) = 2 \cos 70^\circ\)

Now let's look at the denominator, \(\sin 10^\circ \cos 10^\circ\). This is related to the sine double angle formula: \(\sin 2\theta = 2 \sin \theta \cos \theta\). To use this, we can multiply the denominator by 2 and also multiply the whole expression by 2 to keep it balanced (or simply see that \(\sin \theta \cos \theta = \frac{1}{2} \sin 2\theta\)).

Denominator: \(\sin 10^\circ \cos 10^\circ = \frac{1}{2} (2 \sin 10^\circ \cos 10^\circ) = \frac{1}{2} \sin (2 \times 10^\circ) = \frac{1}{2} \sin 20^\circ\)

Now substitute the simplified numerator and denominator back into the expression:

\(\frac{{\cos 10^\circ - \sqrt 3 \sin 10^\circ}}{{\sin 10^\circ \cos 10^\circ}} = \frac{{2 \cos 70^\circ}}{{\frac{1}{2} \sin 20^\circ}}\)

Simplify the fraction:

\(= \frac{{2 \cos 70^\circ}}{1} \times \frac{2}{{\sin 20^\circ}} = \frac{{4 \cos 70^\circ}}{{\sin 20^\circ}}\)

Finally, we can use the complementary angle identity, which states that \(\cos \theta = \sin (90^\circ - \theta)\). Let \(\theta = 70^\circ\).

\(\cos 70^\circ = \sin (90^\circ - 70^\circ) = \sin 20^\circ\)

Substitute this into the expression:

\(= \frac{{4 \sin 20^\circ}}{{\sin 20^\circ}}\)

Since \(\sin 20^\circ \neq 0\), we can cancel \(\sin 20^\circ\):

\(= 4\)

Therefore, the expression \(\frac{1}{{\sin 10^\circ }} - \frac{{\sqrt 3 }}{{\cos 10^\circ }}\) is equal to 4.

Revision Table: Key Trigonometric Identities Used

Additional Information: Manipulating Trigonometric Expressions

Manipulating trigonometric expressions often involves recognizing common patterns and applying appropriate identities. Here are a few points to remember:

• When combining fractions with trigonometric functions in the denominator, find a common denominator first.
• Expressions like \(a \cos \theta + b \sin \theta\) can often be simplified into a single trigonometric function of the form \(R \cos(\theta - \alpha)\) or \(R \sin(\theta + \alpha)\) using the auxiliary angle transformation. The value of R is \(\sqrt{a^2 + b^2}\).
• Keep an eye out for angles like \(30^\circ, 45^\circ, 60^\circ\) as their sine and cosine values (\(\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}\)) often appear as coefficients when using auxiliary angle transformations or other identity manipulations.
• Double angle (\(\sin 2\theta, \cos 2\theta, \tan 2\theta\)) and half-angle identities are frequently useful for simplifying expressions involving multiples or fractions of angles.
• Complementary angle identities (\(\sin \theta = \cos (90^\circ - \theta)\), \(\tan \theta = \cot (90^\circ - \theta)\), etc.) are helpful when you have trigonometric functions of angles that add up to \(90^\circ\), like \(70^\circ\) and \(20^\circ\).

Practicing with various trigonometric problems helps in recognizing which identity or technique to apply.