Trigonometric Identities Quiz

1) Is Sinx + Cosx = 1 an identity?

True

False

Maybe

It depends

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.

Explanation

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.

2) Which of these is the identity of Sin 2x?

Cos^2 x + Sin^2 x

2SinxCosx

SinxCosx + CosxSin x

Sin^2 x

Sin x + Sinx

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

Explanation

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

3) True or False: (tan x + tan y sec x) ÷ (cos x + tan y sin x) = tan(x + y)

True

False

Maybe

It depends

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.

Explanation

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.

4) What is ( Sinx + Cosy )^2 ?

Sin^2x + Cos^2x

1

2SinxCosy

SinxCosy + CosxSiny

Sin^2x + 2SinxCosy + Cos^2y

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.

Explanation

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.

5) Is 2Sec^2 xCos^2 x = 2 an identity?

True

False

Maybe

It depends

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.

Explanation

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.

6) Is Sec2x = 1 an identity? ( 1 - Sinx )( 1 + Sinx)

True

False

Maybe

It depends

Explanation

7) What is Sinx + Cosx + 1 ?
TanxCotx

1

Sinx + Cosx

Cos^2 xSin^2 x

2

______1________ SecxCscx

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

Explanation

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

8) Is Sin^2 x + Sin^2 ( x + y ) + 2SinxCosxSin ( x + y ) = Sin^2 y ?

True

False

Maybe

It depends

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.

Explanation

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.

9) Which of the following is not an identity with Cos2x?

1 - Sin^2 x

2Cosx - 1

Cos^2 x - Sin^2 x

1 - 2Sin^2 x

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).

Explanation

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).

10) Which of the following is the correct form of the Pythagorean identity in trigonometry?

Sin2x+cos2x=1

Tan2x+cot2x=1

Sec2x+csc2x=1

Sin2x+tan2x=1

Explanation