Solving Trigonometric Equations: The Value of cot θ Given sin θ cos θ √2 cos 90 - θ

Trigonometry is a fascinating branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key concepts in trigonometry is the manipulation of trigonometric identities to simplify equations and solve for unknown variables. In this article, we will explore a specific problem involving the cotangent of an angle, given the equation sin θ cos θ √2 cos 90 - θ. We will use various trigonometric identities to simplify and solve for cot θ.

Understanding the Given Equation

The given equation is: sin θ cos θ 2 cos 90 #x00B0; - θ

Using a co-function identity, we know that:

cos 90 #x00B0; - θ sin θ

Substituting this identity into the given equation, we get:

sin θ cos θ 2 sin θ

Dividing both sides by sin θ (assuming sin θ ≠ 0):

cos θ 2 - 1 sin θ

Expressing cot θ in Terms of the Given Identity

We know that:

cot θ cos θ sin θ

Substituting the expression for cos θ from above:

cot θ 2 - 1 sin θ sin θ

This simplifies to:

cot θ 2 - 1

Additional Perspective

Another way to approach this problem is to let:

θ A

Then, the equation becomes:

sin A cos A 90 - A

or

or

cos A square root of 2 - 1 times sin A

or

cos A sin A square root of 2 - 1

or

cot A square root of 2 - 1

Since θ A, we have:

cot θ square root of 2 - 1

This confirms our previous result.

Conclusion

The value of cot θ, given the equation sin θ cos θ √2 cos 90 - θ, is:

square root of 2 - 1

Thus, the final answer is:

cot θ 2 - 1