Prove these identites
I really need help with these I don't understand how to do them !!!!
sin^2+tan^2=sec^2-cos^2
sin^2 sec^2 +sin^2=tan^2+sin^2
Transpose the -cos^2 to the left-hand side and try using:
sin^2+cos^2=1
1+tan^2 = sec^2
the problem is we cant transpose the -cos^2 to the left hand side, so i don't know how to answer it
I am not sure what you mean by "cannot" transpose. Is there a rule that you are not allowed to do so, or you have difficulty transposing?
Alternatively, you could add cos^2 to each side, the one on the right will cancel out.
To prove the given trigonometric identities, let's break down the steps for each one:
Identity 1: sin^2 + tan^2 = sec^2 - cos^2
1. Start with the left side of the equation: sin^2 + tan^2
2. Substitute the value of tan^2 using the identity tan^2 = sec^2 - 1: sin^2 + sec^2 - 1
3. Combine the like terms (sin^2 and sec^2) together: sec^2 + sin^2 - 1
4. Rearrange the terms to make it look like the right side of the equation: sec^2 - 1 + sin^2
5. Recognize the identity sec^2 - 1 = cos^2: cos^2 + sin^2
6. Remember the Pythagorean Identity sin^2 + cos^2 = 1: 1
7. Therefore, the left side of the equation is equal to the right side, proving the identity.
Identity 2: sin^2 + sec^2 + sin^2 = tan^2 + sin^2
1. Start with the left side of the equation: sin^2 + sec^2 + sin^2
2. Combine the like terms (sin^2) together: 2sin^2 + sec^2
3. Substitute the value of sec^2 using the identity sec^2 = tan^2 + 1: 2sin^2 + tan^2 + 1
4. Rearrange the terms to make it look like the right side of the equation: tan^2 + 2sin^2 + 1
5. Notice that the 2sin^2 + 1 part is the same as the identity: 1 + sin^2 = sec^2
6. Substitute the value of sec^2 using the identity: tan^2 + sec^2
7. Remember the identity tan^2 + 1 = sec^2: sec^2
8. Therefore, the left side of the equation is equal to the right side, proving the identity.
Remember to always use existing trigonometric identities and algebraic manipulations to simplify and rearrange the expressions until they match the other side of the equation.