Trigonometry

Step 1: Determine the value of .

From , we recognize this as a standard trigonometric value. The angle that satisfies this is:

Step 2: Use the given .

Substitute :

Recognize that is the tangent of :

Step 3: Solve for .

Since , we equate the angles:

Solve for :

Final Answer: The value of is , which corresponds to option .

Step 1: Recall the double-angle identity for tangent.

The double-angle identity for tangent is:

Here, , so:

Step 2: Simplify .

From trigonometric values, we know:

The expression simplifies to , which corresponds to option .

Final Answer: The value of is , which corresponds to option .

Step 1: Use the identity for cotangent.

The cotangent of an angle is the reciprocal of the tangent:

Substitute this into the given expression:

Simplify the division:

Step 2: Use the complementary angle identity.

Since , we know that:

Thus, the expression becomes:

Using the identity :

Final Answer: The value of is .

We are tasked with simplifying .

Step 1: Use trigonometric identities for complementary angles.

Substituting these into the expression:

Step 2: Simplify using a double-angle identity.

The double-angle identity for sine is:

Rearranging, we get:

Step 3: Express in terms of .

The identity for in terms of is:

Substitute this into :

Simplify:

Final Answer: The value of is , which corresponds to option .

The given expression is: This product involves tangent values for angles from to . Notice that: Thus, for each pair of terms in the product, such as and , we have: Similarly: This pattern continues for all pairs, leading to a total product of 1. Therefore, the value of the expression is:

The correct option is (B):

Given:

Step 1: Express in terms of sin and cos

Step 2: Assign standard values

For , we know:

Step 3: Compute sec θ

Final Answer: 2

Given:

Step 1: Express in terms of tan

So,

Step 2: Square both sides

Expanding using the identity ,

Step 3: Solve for

Final Answer: 23

Given Expression:

Step 1: Use the Cosine Addition Identity

We know that: Comparing with our expression, we see that:

Step 2: Compute

Final Answer: 0

Given:

Step 1: Use the identity

Applying this identity:

Step 2: Multiply the expressions

Final Answer: 1

We are given that .
We know that and .
Substituting these values into the equation, we get $ \frac{a^4}{b^4} \frac{a^4}{b^4} = 64$.

We are given that .

We know that .

So, we can rewrite .

Similarly, .

Therefore, we have $ 1 = \tan(A+30^\circ) 45^\circ A = 15^\circ$.