1.
Which of these is the identity of Sin 2x?
Correct Answer
B. 2SinxCosx
Explanation
Sin 2x = Sin ( x + y )
= Sin ( x + x )
= SinxCosx + CosxSinx
= 2SinxCosx
2.
Is Sec2x = 1 an identity? ( 1 - Sinx )( 1 + Sinx)
Correct Answer
A. True
Explanation
The given expression is (1 - Sinx)(1 + Sinx). This expression can be simplified to 1 - Sin^2x using the identity (a + b)(a - b) = a^2 - b^2. Since Sin^2x is equal to 1 - Cos^2x, the expression can be further simplified to Cos^2x. Therefore, the expression is equal to Cos^2x, which is an identity. Therefore, the correct answer is true.
3.
True or false, Tanx + TanySecx = Tan ( x + y ) ? Cosx + TanySinx
Correct Answer
B. 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.
Is Sin^2
x + Sin^2
( x + y ) + 2SinxCosxSin ( x + y ) = Sin^2 y ?
Correct Answer
A. 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.
5.
What is ( Sinx + Cosy )^2 ?
Correct Answer
E. Sin^2x + 2SinxCosy + Cos^2y
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.
6.
What is Cos2x + Sec2x
?
Correct Answer
D. Tan^2 2x
Explanation
The given expression is equal to Tan^2 2x. This can be derived using trigonometric identities. Cos^2x can be written as 1 - Sin^2x, and Sec^2x can be written as 1 + Tan^2x. By substituting these values into the expression, we get 1 - Sin^2x + 1 + Tan^2x. Simplifying further, we get 2 - Sin^2x + Tan^2x. Using the identity Tan^2x = Sin^2x / Cos^2x, we can rewrite the expression as 2 - Sin^2x + Sin^2x / Cos^2x. Combining like terms, we get 2 + Sin^2x / Cos^2x. Finally, using the identity Sin^2x / Cos^2x = Tan^2x, we arrive at Tan^2 2x.
7.
What is Sinx + Cosx + 1 ? TanxCotx
Correct Answer
B. Sinx + Cosx
Explanation
The given expression is Sinx + Cosx. This is the correct answer because it matches with the expression provided in the question.
8.
Is 2Sec^2 xCos^2 x = 2
an identity?
Correct Answer
A. True
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.
9.
Which of the following is not an identity with Cos2x?
Correct Answer
A. 1 - Sin^2 x
Explanation
The given expression is 1 - Sin^2 x. This can be simplified using the identity Cos^2 x = 1 - Sin^2 x. By substituting this identity into the expression, we get 1 - Sin^2 x = Cos^2 x, which is an identity with Cos2x. Therefore, the correct answer is 1 - Sin^2 x, as it is not an identity with Cos2x.
10.
Is Sinx + Cosx = 1 an identity?
Correct Answer
B. False
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.