Please I need help.Show full working out. (a) sin x cot x= cos x (b) (1 + tan theta)^2 + (1- tan theta)^2 = 2sec^2theta
since cot = cos/sin,
sin cot = sin * cos/sin = cos
expand the two tan expressions and recall that
sec^2 = 1 + tan^2
Sure, I can help you with that. Let's solve each equation step by step and show the working out.
(a) sin x cot x = cos x:
To solve this equation, we will use trigonometric identities and algebraic manipulations to simplify the expression.
First, rewrite cot x as 1/tan x since cotangent is the reciprocal of tangent: sin x (1/tan x) = cos x.
Divide both sides of the equation by sin x: 1/tan x = cos x/sin x.
Next, simplify the right side of the equation using the identity: cos x/sin x = cot x.
Now, we have 1/tan x = cot x.
Since both sides of the equation are in terms of cot x, we can rewrite the equation as: 1 = tan x cot x.
Using the trigonometric identity tan x cot x = 1, we find that both sides are equal.
Therefore, the solution to the equation sin x cot x = cos x is x = nπ, where n is an integer.
(b) (1 + tan θ)^2 + (1 - tan θ)^2 = 2sec^2θ:
To solve this equation, we will use trigonometric identities and expand the expression on the left side.
First, expand the square terms: (1 + tan θ)^2 = 1 + 2tan θ + tan^2 θ and (1 - tan θ)^2 = 1 - 2tan θ + tan^2 θ.
Now, substitute these expanded terms back into the original equation: (1 + 2tan θ + tan^2 θ) + (1 - 2tan θ + tan^2 θ) = 2sec^2θ.
Simplify the equation by combining like terms: 2 + 2tan^2 θ = 2sec^2θ.
Next, use the trigonometric identity sec^2θ = 1 + tan^2 θ.
Substitute this identity into the equation: 2 + 2tan^2 θ = 2(1 + tan^2 θ).
Simplify further: 2 + 2tan^2 θ = 2 + 2tan^2 θ.
Since both sides of the equation are equal, it means that the equation is true for all values of θ.
Therefore, the solution to the equation (1 + tan θ)^2 + (1 - tan θ)^2 = 2sec^2θ is θ belongs to all real numbers.