Trigonometric Identities Quiz

The equation Sinx + Cosx = 1 is not an identity because it is not true for all values of x. There are certain values of x for which the equation holds true, but there are also values for which it does not hold true. Therefore, it cannot be considered an identity.

Sin 2x = Sin ( x + y )
= Sin ( x + x )
= SinxCosx + CosxSinx
= 2SinxCosx

The given equation is "Tanx + TanySecx = Tan(x + y)". However, the correct equation should be "Tanx + TanySecx = Tan(x + y) + Cosx + TanySinx". Therefore, the answer is false.

The given expression is (Sin x + Cos y)^2. When we expand this expression, we get Sin^2 x + 2Sin x Cos y + Cos^2 y. Therefore, the correct answer is Sin^2 x + 2Sin x Cos y + Cos^2 y.

The given expression, 2Sec^2 xCos^2 x, simplifies to 2, which is a constant value regardless of the value of x. Therefore, the expression is an identity, meaning it holds true for all values of x.

The given expression is Sinx + Cosx. This is the correct answer because it matches with the expression provided in the question.

The given equation is true. This can be proven by using the trigonometric identity: Sin^2(x) + Cos^2(x) = 1. By substituting Sin^2(x) with (1 - Cos^2(x)) in the equation, we can simplify it to: (1 - Cos^2(x)) + Sin^2(x + y) + 2Sin(x)Cos(x)Sin(x + y) = Sin^2(y). Simplifying further, we get: 1 + Sin^2(x + y) + 2Sin(x)Cos(x)Sin(x + y) = Sin^2(y). Since 1 + Sin^2(x + y) is equal to Sin^2(y) according to the trigonometric identity, the equation is true.

The formula cos(2x) = cos²x − sin²x is the double-angle identity derived from the basic cosine addition formula:

cos(a + b) = cos a cos b − sin a sin b

So setting a = b = x gives:

cos(2x) = cos²x − sin²x

From this, using the Pythagorean identity sin²x = 1 − cos²x, or vice versa, you derive the other two forms:

cos(2x) = 2cos²x − 1

cos(2x) = 1 − 2sin²x

But 2sinx cosx = sin(2x), which belongs to sine double angle, not cosine. Hence, it's not a valid identity for cos(2x).