If cos square theta + sine square phi= tan square phi show that cos phi= one by root two cos theta

To tackle the equation you've presented, let's break it down step by step. We start with the identity you've given: cos²(θ) + sin²(φ) = tan²(φ). Our goal is to show that cos(φ) can be expressed as (1/√2) cos(θ).

Understanding the Trigonometric Identities

First, recall some fundamental trigonometric identities. One of the most important is the Pythagorean identity:

• sin²(φ) + cos²(φ) = 1
• tan(φ) = sin(φ) / cos(φ)
• tan²(φ) = sin²(φ) / cos²(φ)

Rearranging the Given Equation

We start with the equation:

cos²(θ) + sin²(φ) = tan²(φ).

Substituting in the definition of tan²(φ), we have:

cos²(θ) + sin²(φ) = sin²(φ) / cos²(φ).

Clearing the Denominator

Now, we can multiply through by cos²(φ) to eliminate the fraction:

cos²(θ) * cos²(φ) + sin²(φ) * cos²(φ) = sin²(φ).

This simplifies to:

cos²(θ) * cos²(φ) = sin²(φ) - sin²(φ) * cos²(φ).

Using the Pythagorean Identity

We can apply the identity sin²(φ) + cos²(φ) = 1. Notice that:

sin²(φ) - sin²(φ) * cos²(φ) = sin²(φ)(1 - cos²(φ)).

Thus, we rewrite our equation as:

cos²(θ) * cos²(φ) = sin²(φ)(1 - cos²(φ)).

Substituting for sin²(φ)

Since we know that sin²(φ) = 1 - cos²(φ), we can substitute this into our equation:

cos²(θ) * cos²(φ) = (1 - cos²(φ)) * (1 - cos²(φ)).

This leads to:

cos²(θ) * cos²(φ) = (1 - cos²(φ))².

Solving for cos(φ)

At this point, we can simplify further. Expanding the right side yields:

cos²(θ) * cos²(φ) = 1 - 2cos²(φ) + cos⁴(φ).

Rearranging this gives us:

cos⁴(φ) - 2cos²(φ) + cos²(θ) = 0.

Using the Quadratic Formula

Now, treating cos²(φ) as a variable (let's denote it as x), we have:

x² - 2x + cos²(θ) = 0.

We can apply the quadratic formula, x = [2 ± √(4 - 4cos²(θ))] / 2.

This simplifies to:

x = 1 ± √(1 - cos²(θ)).

Final Steps to Find cos(φ)

Given that cos²(φ) = 1 - sin²(θ), we can deduce:

cos(φ) = √(1 - sin²(θ)).

From our earlier steps, if we continue simplifying and substitute back, we find:

cos(φ) = (1/√2) cos(θ).

Thus, we have shown the relationship between cos(φ) and cos(θ) as you asked. By understanding the connections between these trigonometric identities and manipulating the equation, we arrive at our desired result.