Solving Trigonometry Expression: Step-by-Step

Let's find the value of the given trigonometry expression: \(cosec(60° + A) - sec(30° - A) +\frac{cosec49^{\circ}}{sec 41^{\circ}}\).

To solve this, we will analyze and simplify each part of the expression using trigonometric identities, specifically focusing on complementary angles.

Understanding Complementary Angle Identities

Two angles are called complementary if their sum is \(90^\circ\). There are standard trigonometric identities relating functions of complementary angles:

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

Simplifying the First Part: \(cosec(60° + A) - sec(30° - A)\)

Consider the angles \((60° + A)\) and \((30° - A)\). Let's check their sum:

\((60° + A) + (30° - A) = 60° + A + 30° - A = 90°\)

Since the sum of the angles is \(90^\circ\), they are complementary angles.

We can use the identity \(sec(90^\circ - \theta) = cosec(\theta)\).

Let \(\theta = 60° + A\). Then \(90^\circ - \theta = 90^\circ - (60° + A) = 90° - 60° - A = 30° - A\).

So, \(sec(30° - A) = sec(90^\circ - (60° + A))\).

Using the identity, \(sec(90^\circ - (60° + A)) = cosec(60° + A)\).

Thus, \(sec(30° - A) = cosec(60° + A)\).

Now substitute this back into the first part of the expression:

\(cosec(60° + A) - sec(30° - A) = cosec(60° + A) - cosec(60° + A) = 0\)

Simplifying the Second Part: \(\frac{cosec49^{\circ}}{sec 41^{\circ}}\)

Consider the angles \(49^\circ\) and \(41^\circ\). Let's check their sum:

\(49^\circ + 41^\circ = 90^\circ\)

Since the sum is \(90^\circ\), these angles are also complementary.

We can use the identity \(sec(90^\circ - \theta) = cosec(\theta)\).

Let \(\theta = 49^\circ\). Then \(90^\circ - \theta = 90^\circ - 49^\circ = 41^\circ\).

So, \(sec(41^\circ) = sec(90^\circ - 49^\circ)\).

Using the identity, \(sec(90^\circ - 49^\circ) = cosec(49^\circ)\).

Thus, \(sec(41^\circ) = cosec(49^\circ)\).

Now substitute this into the second part of the expression:

\(\frac{cosec49^{\circ}}{sec 41^{\circ}} = \frac{cosec49^{\circ}}{cosec49^{\circ}}\)

Assuming \(cosec49^{\circ} \neq 0\) (which is true as \(sin49^{\circ} \neq 0\)), the fraction simplifies to 1.

\(\frac{cosec49^{\circ}}{sec 41^{\circ}} = 1\)

Combining the Simplified Parts

The original expression is the sum of the two simplified parts:

\((cosec(60° + A) - sec(30° - A)) + (\frac{cosec49^{\circ}}{sec 41^{\circ}})\)

Substitute the simplified values:

\(0 + 1 = 1\)

Final Value of the Expression

The value of the given trigonometry expression is \(1\).

Revision Table: Key Trigonometry Concepts

Additional Information on Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every single value of the occurring variables where both sides of the equality are defined. These identities are fundamental in trigonometry and are used to simplify expressions, solve equations, and prove other relationships.

The complementary angle identities are special cases of the angle addition/subtraction formulas, but they are often remembered separately due to their frequent use in simplifying expressions involving angles like \((90^\circ - \theta)\). They help in converting trigonometric ratios of angles greater than \(45^\circ\) to ratios of angles less than \(45^\circ\) or in relating co-functions (like sin and cos, tan and cot, sec and cosec).

Mastering these identities is crucial for solving trigonometry problems, especially those encountered in exams. Practice applying these identities to various problems to become proficient in simplifying complex trigonometric expressions.