Trigonometry

Let .
This means that .
Similarly, let ,
so .
Now, consider the sum . Using the identity: we have: Now, the original expression becomes:

The correct option is (E) :

We are given that: To find , we will use the Pythagorean identity: Step 1: Substitute the given value of We know that , so: Step 2: Solve for Subtract from both sides: Step 3: Take the square root of both sides Since , we have: Step 4: Use the condition Since and , we must choose the negative value for because . Thus,

The correct option is (D) :

Given:

• Equation:
• Real values and

Step 1: Let and . Rewrite the equation:

Step 2: Take cube roots of both sides:

Step 3: Apply componendo-dividendo:

Step 4: Similarly, apply componendo-dividendo to the left side:

Step 5: Let . Then:

Step 6: Substitute back for : This matches the original equation, confirming our approach.

Step 7: Solve for in terms of :

Step 8: Compute :

Conclusion: The correct expression is .

Given:

• Expression to evaluate:

Step 1: Rewrite trigonometric functions in terms of sine and cosine:

Step 2: Substitute known values:

Step 3: Rewrite the original expression:

Step 4: Combine the terms:

Step 5: Recognize the numerator as a trigonometric identity:

Step 6: Simplify the denominator using double-angle identity:

Step 7: Final simplification:

The correct answer is (D) 4.

Given:

1 = 2k


So, the correct option is (B) : 2.

Given:

Step 1: Use trigonometric identities:

Step 2: Substitute into the equation:

Step 3: Apply triangle identity:

Step 4: Solve for the product:

Given:
Here , x is the real number

⇒

⇒

⇒

⇒

⇒

⇒

Hence ,the correct option is

Step 1: Differentiate the function : Using the double-angle identity:

Step 2: Evaluate the derivative at :

Step 3: Set the derivative equal to the given value:

Given:

We are asked to find the value of:

tan(cos^-1(-24/25))

Let θ = cos^-1(-24/25), so cos(θ) = -24/25.

We need to find tan(θ), which is given by:

tan(θ) = sin(θ) / cos(θ)

Using the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

Substitute cos(θ) = -24/25:

sin²(θ) + (-24/25)² = 1

sin²(θ) + 576/625 = 1

sin²(θ) = 1 - 576/625 = 625/625 - 576/625 = 49/625

sin(θ) = 7/25

Now, calculate tan(θ):

tan(θ) = sin(θ) / cos(θ) = (7/25) / (-24/25) = 7 / -24 = -7/24

Thus, the value of tan(cos^-1(-24/25)) is -7/24.

Step 1: Use the Identity for Sum of Two Arctangents Using the identity: Setting and : Step 2: Use the Cotangent-Arctangent Identity Using the identity: for , we get: Since we already computed: Adding :
Final Answer:

Step 1: Square Both Sides Using , we get: Step 2: Compute Using the identity:
Final Answer:

Step 1: Rewrite in Terms of Half-Angle Identity Using the identity: Using the half-angle identity:
Final Answer:

Step 1: Express in Terms of Sine and Cosine Step 2: Compute the Given Expression Step 3: Factorize Since , we get:
Final Answer:

Step 1: Define the Terms Let: Using the identity: Step 2: Compute Using the double angle identity: Step 3: Compute Using the identity: After simplification,
Final Answer:

We are given: We are asked to find . First, substitute the expressions for and into : Now simplify the expression by multiplying the numerator and the denominator: Simplify the terms: Using the identity , we find the correct simplified expression to be:
Thus, the correct answer is .