Evaluating the Trigonometric Expression: 5 sin 14° sec 76° + 3 cot 15° cot 75° + 2 tan 45°

Let's evaluate the given trigonometric expression step by step. The expression is:

$$5 \sin 14^\circ \sec 76^\circ + 3 \cot 15^\circ \cot 75^\circ + 2 \tan 45^\circ$$

We will simplify each term individually using trigonometric identities, particularly the complementary angle identities and reciprocal identities.

Simplifying the First Term: 5 sin 14° sec 76°

The first term is $5 \sin 14^\circ \sec 76^\circ$.

• We notice that $14^\circ + 76^\circ = 90^\circ$. These are complementary angles.
• Using the complementary angle identity $\sec (90^\circ - \theta) = \csc \theta$, we can write $\sec 76^\circ$ as $\sec (90^\circ - 14^\circ)$.
• So, $\sec 76^\circ = \csc 14^\circ$.
• Now, using the reciprocal identity $\csc \theta = \frac{1}{\sin \theta}$, we have $\csc 14^\circ = \frac{1}{\sin 14^\circ}$.
• Substitute this back into the first term:
• $$5 \sin 14^\circ \sec 76^\circ = 5 \sin 14^\circ \left(\frac{1}{\sin 14^\circ}\right)$$
• Assuming $\sin 14^\circ \neq 0$ (which is true as $14^\circ$ is in the first quadrant), we can cancel out $\sin 14^\circ$.
• The first term simplifies to $5 \times 1 = 5$.

Simplifying the Second Term: 3 cot 15° cot 75°

The second term is $3 \cot 15^\circ \cot 75^\circ$.

• We notice that $15^\circ + 75^\circ = 90^\circ$. These are also complementary angles.
• Using the complementary angle identity $\cot (90^\circ - \theta) = \tan \theta$, we can write $\cot 75^\circ$ as $\cot (90^\circ - 15^\circ)$.
• So, $\cot 75^\circ = \tan 15^\circ$.
• Now, using the reciprocal identity $\tan \theta = \frac{1}{\cot \theta}$, we have $\tan 15^\circ = \frac{1}{\cot 15^\circ}$.
• Substitute this back into the second term:
• $$3 \cot 15^\circ \cot 75^\circ = 3 \cot 15^\circ (\tan 15^\circ)$$
• $$3 \cot 15^\circ (\tan 15^\circ) = 3 \cot 15^\circ \left(\frac{1}{\cot 15^\circ}\right)$$
• Assuming $\cot 15^\circ \neq 0$ (which is true for $15^\circ$), we can cancel out $\cot 15^\circ$.
• The second term simplifies to $3 \times 1 = 3$.

Simplifying the Third Term: 2 tan 45°

The third term is $2 \tan 45^\circ$.

• We know the standard value of $\tan 45^\circ$.
• The value of $\tan 45^\circ = 1$.
• Substitute this value into the third term:
• $$2 \tan 45^\circ = 2 \times 1$$
• The third term simplifies to $2$.

Combining the Simplified Terms

Now, we add the simplified values of the three terms to find the value of the entire expression.

Expression value = (Value of First Term) + (Value of Second Term) + (Value of Third Term)

Expression value = $5 + 3 + 2$

Expression value = $10$

Thus, the value of the given expression $5 \sin 14^\circ \sec 76^\circ + 3 \cot 15^\circ \cot 75^\circ + 2 \tan 45^\circ$ is 10.

Revision Table: Key Trigonometric Concepts

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all valid values of the variables. They are fundamental tools for simplifying expressions, solving equations, and proving other identities.

• Reciprocal Identities: Relate the six trigonometric functions:
  • $\csc \theta = \frac{1}{\sin \theta}$
  • $\sec \theta = \frac{1}{\cos \theta}$
  • $\cot \theta = \frac{1}{\tan \theta}$
  • $\sin \theta = \frac{1}{\csc \theta}$
  • $\cos \theta = \frac{1}{\sec \theta}$
  • $\tan \theta = \frac{1}{\cot \theta}$
• Quotient Identities: Express tangent and cotangent in terms of sine and cosine:
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• Pythagorean Identities: Derived from the Pythagorean theorem:
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$
• Complementary Angle Identities: Relate trigonometric functions of an angle to those of its complement ($90^\circ - \theta$):
  • $\sin (90^\circ - \theta) = \cos \theta$
  • $\cos (90^\circ - \theta) = \sin \theta$
  • $\tan (90^\circ - \theta) = \cot \theta$
  • $\cot (90^\circ - \theta) = \tan \theta$
  • $\sec (90^\circ - \theta) = \csc \theta$
  • $\csc (90^\circ - \theta) = \sec \theta$

Mastering these identities is crucial for solving problems involving trigonometric expressions and equations.