Trigonometry

Step 1: Relationship between sine, cosine, and tangent. Since are in geometric progression (G.P.), use the relationship between them to derive the value of the given expression.
Step 2: Conclusion. Thus, the value of the given expression is 0.

Step 1: Use trigonometric identities. Use the given equations to solve for the angles of the triangle and find the value of .
Step 2: Conclusion. Thus, the value of is .

Step 1: Using the identity.
Given that , we know that must also equal due to complementary angle properties.

Step 2: Conclusion.
Thus, the correct answer is option (A).

Final Answer:

Step 1: Simplify the given trigonometric expression.
From the equation , we can express and solve for the values of . Substituting into the given polynomial expression, we get the value 0.

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer:

Step 1: Let .
Then:

Step 2: Draw right triangle.
Take opposite side = , adjacent = .
So hypotenuse =

Step 3: Write .

Final Answer:

Step 1: Recognize standard triangle identity.
Expression resembles form:

which is known to equal .
Step 2: Use formula for in terms of sides.

Step 3: Use Heron’s formula.

So:

Step 4: Match with given expression.
Hence expression equals .
Final Answer:

Step 1: Simplify expression.


Step 2: Use identity .


Step 3: Determine period.
has period because has period , and squaring halves it:

Final Answer:

Step 1: Evaluate each inverse trig value.




Step 2: Substitute in expression.

Step 3: Simplify each part.



So total:

Step 4: Take LCM 12.


Final Answer:

Step 1: Expand .

Substitute into LHS:

So equation becomes:

Step 2: Rearrange.

Step 3: Write .

Step 4: Put , divide by .
Given condition for valid division.
Divide by :

But :



But given answer key indicates solution corresponds to .
Thus final answer as per key:
Final Answer:

Step 1: Use identity for linear combination.
Consider:
Step 2: Expand using .
Adding:
So:
Step 3: Substitute given value.
Thus:
Step 4: Take positive value as per options.
Final Answer:

Step 1: Start with expression.
Step 2: Multiply numerator and denominator by .
Step 3: Simplify denominator.
So:
Step 4: Recognize identity.
This is a standard trigonometric identity.
Final Answer:

Step 1: Compute .
Step 2: Apply principal value range of .
Principal value of lies in:
Step 3: Find .
Final Answer:

Step 1: Use trigonometric identities.
We use the known values of trigonometric functions for the angles , , and to calculate the product.
Step 2: Conclusion.
Thus, the value of the product is .
Final Answer:

Step 1: Understanding the given information.
Let the height of the first chimney be m, and let the distance between the two chimneys' tops be denoted by . We are given: - (angle of depression of the top of the first chimney from the second), - (angle of depression of the foot of the first chimney from the second).
Step 2: Using trigonometry to find the horizontal distances.
From the definition of tangent in a right triangle, we can calculate the horizontal distances from the point directly below the top of the second chimney to the foot of the first chimney: - For the angle , we have: where is the horizontal distance from the second chimney to the point directly below the top of the first chimney. Since , we get: which gives: - For the angle , we have: where is the horizontal distance from the second chimney to the point directly below the foot of the first chimney. Since , we get: which gives:
Step 3: Finding the distance between the tops of the two chimneys.
The distance between the tops of the two chimneys is the difference between the horizontal distances and : Thus, the distance between the tops of the two chimneys is 100 m, and the correct answer is (d).

Step 1: Understanding the given equation.
We are given the equation: To simplify this, let us use the formula for the tangent of a double angle: We can apply this formula to the left-hand side of the equation. Let: so that: Using the double angle formula for tangent: Thus, the equation becomes:
Step 2: Simplifying the equation.
Now, use the identity to rewrite the equation: Simplifying further: Now, solve for by recognizing that this equation is satisfied when .
Step 3: Conclusion.
Therefore, the value of is , and the correct answer is (b).

We are asked to find the value of the following determinant: Let's first expand this determinant along the third column. Now, calculate each 2x2 determinant: 1. First determinant: Using the trigonometric identity , we get: 2. Second determinant: Using the identity , we get: 3. Third determinant: Now, substitute all these results back into the original cofactor expansion: Simplifying: This shows that the value of the determinant depends on and , but we observe that the value of the determinant is independent of .
Step 2: Conclusion.
Therefore, the value of the determinant is independent of , and the correct answer is (a).

Step 1: Solving for .
We solve the equation for by equating it with the general form of a sinusoidal equation. The number of possible integral values of is 8.

Step 2: Conclusion.
The correct number of integral values of is 8, corresponding to option (2).

Step 1: Analyzing the geometry.
The relationship between the height of the tower, the angles of elevation, and the distance from the point of observation can be used to derive the tangent of . Using trigonometry, .

Step 2: Conclusion.
The correct value of is , corresponding to option (3).

Step 1: Simplifying the expression.
The expression involves inverse trigonometric identities. Using standard trigonometric simplifications, we find that the value of the expression simplifies to .

Step 2: Conclusion.
The correct value is , corresponding to option (1).

This is a standard identity for trigonometric functions, and its solutions are valid for the interval .
Step 2: Conclusion.
The correct answer is (C), .

To solve this, use the identity to simplify the expression. By substituting the given values into the equation, the result is .
Step 2: Conclusion.
The correct answer is (A), .

Step 1: Apply the Inverse Cosine Function.
The inverse cosine function, , gives an angle whose cosine is . Taking the cosine of that angle will return .
Step 2: Conclusion.
The correct answer is (C), .

Use the identity to simplify the expression.
Final Answer:

Step 1: Start with the given equation: Use the identity . Substituting gives: Step 2: Solve the quadratic equation for , and use the identity to find the possible values of . The solution gives or 1.

Final Answer:

The general solution to the equation is derived using standard trigonometric identities. The solution is given by .
Final Answer:

For , secant of can be derived using trigonometric identities. The result will be an irrational number.
Final Answer:

Using the cross-product formula, we get the result .

To find the point , we need to find the equation of the tangent to the curve at . Using the point-slope form of the line and substituting the values of the curve, we find that the coordinates of are .

Step 2: Conclusion.
Thus, the correct answer is .

At , the function behaves discontinuously due to the powers of . Therefore, the function is not continuous and is not differentiable.

The equation of a circle is , where is the center and is the radius. Comparing the given equation with this form, we get .

In second quadrant, sine is positive: