If \(\cos \theta+\sec \theta=2\) then \(\cos ^{2012} \theta+\sec ^{2012} \theta=\) (a) \(2^{2012}\) (b) \(2^{2013}\) (c) 2 (d) 0

Understanding the relationship between cosine (\text{cos}) and secant (\text{sec}) functions is critical when dealing with trigonometric equations. Remember that secant is the reciprocal of cosine, which is mathematically represented as \text{sec} \theta = \frac{1}{\cos \theta}.

This reciprocal relationship makes it possible to convert between the two functions. If we have an equation like \(\text{cos} \theta + \text{sec} \theta = 2\), we can easily substitute \text{sec} \theta with \frac{1}{\text{cos} \theta} to simplify and solve for \text{cos} \theta. Using this substitution technique allows us to avoid working directly with secant, which can sometimes complicate the problem-solving process.

When dealing with trigonometric identities, we often need to transform the original equation to make it more solvable. Squaring both sides is a common method used to eliminate radicals or fractions. But, be cautious—this operation can introduce extraneous solutions.

For instance, if we square \(\text{cos} \theta + \frac{1}{\text{cos} \theta} = 2\), we must expand and simplify carefully, using the identity \(\text{cos}^2 \theta + 2 + \frac{1}{\text{cos}^2 \theta} = 4\).

• Remember that when squaring the sum, you should apply the formula \((a+b)^2 = a^2 + 2ab + b^2\) to get the correct expansion.
• Ensure to verify any potential solutions since not all solutions of the squared equation will satisfy the initial equation.

Clearly understanding how to manage and manipulate squared identities can prevent miscalculations and errors in solutions.

When you're solving quadratic equations in trigonometry, you're often dealing with equations where the unknown variable is a trigonometric function, such as \(\cos^2 \theta\) or \(\sin^2 \theta\).

After squaring an equation and rearranging it into the familiar quadratic form \(ax^2 + bx + c = 0\), you can proceed as you usually would in algebra. The quadratic formula or factoring are common methods used to solve for \(x\), which in trigonometric problems is usually a squared trig function.

Remembering Simple Solutions

• Some quadratic trigonometric equations have simple solutions such as \(x = 1\) or \(x = 0\) when they're perfect squares like \(x^2 - 2x + 1 = 0\).
• After solving the quadratic equation, you must interpret the solution in the context of the trigonometric identity involved, often reverting back to the original trigonometric function.

Being thorough with every step in the process ensures that you do not only solve the equation correctly but also appropriately apply the solution within the scope of trigonometry.