Cos square theta divided by 1-tan theta +sin cube theta divided by sin theta -cos theta = 1+sin theta *cos theta
cos^2 θ /(1 - tanθ) + sin^3 θ/(sinθ - cosθ) = 1 + sinθ cosθ
LS = cos^2 θ /(1 - sinθ/cosθ) + sin^3 θ/(sinθ - cosθ)
= cos^2 θ /((cosθ - sinθ)/cosθ) + sin^3 θ/(sinθ - cosθ)
= cos^3 θ /(cosθ - sinθ) - sin^3 θ/(cosθ - sinθ)
= (cos^3 θ - sin^3 θ)/(cosθ - sinθ) <---- I see the difference of cubes
= (cosθ - sinθ)(cos^2 θ + sinθcosθ + sin^2 θ)/(cosθ - sinθ)
= (cos^2 θ + sinθcosθ + sin^2 θ)
= 1 + sinθcosθ
= RS
wheeww!!
If your question mean.
Prove:
cos² ( θ ) / [ 1 - tan ( θ ) ] + sin³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] = 1 + sin ( θ ) ∙ cos ( θ )
then
cos² ( θ ) / [ cos ( θ ) / cos ( θ ) - sin ( θ ) / cos ( θ ) ] + sin³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] =
cos² ( θ ) / [ cos ( θ ) - sin ( θ ) ] / cos ( θ ) + sin³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] =
cos³ ( θ ) / [ cos ( θ ) - sin ( θ ) ] + sin³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] =
- cos³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] + sin³ ( θ ) / [ sin ( θ ) - cos ( θ ) ] =
[ sin³ ( θ ) - cos³ ( θ ) ] / [ sin ( θ ) - cos ( θ ) ] = 1 + sin ( θ ) ∙ cos ( θ ) =
___________________________
Since:
a³ - b³ = ( a - b ) ∙ ( a² + a ∙ b + b² )
___________________________
[ sin ( θ ) - cos ( θ ) ] ∙ [ sin² ( θ ) + sin ( θ ) ∙ cos ( θ ) + cos ( θ )² ] / [ sin ( θ ) - cos ( θ ) ] =
sin² ( θ ) + sin ( θ ) ∙ cos ( θ ) + cos ( θ )² =
sin² ( θ ) + cos ( θ )² + sin ( θ ) ∙ cos ( θ ) =
1 + sin ( θ ) ∙ cos ( θ )
By the way:
1 + sin ( θ ) ∙ cos ( θ ) = 1 + ( 1 / 2 ) sin ( 2 θ )
To prove the equation:
cos^2(theta) / (1 - tan(theta)) + sin^3(theta) / (sin(theta) - cos(theta)) = 1 + sin(theta) * cos(theta)
Let's begin by simplifying each term on the left side of the equation.
1. Simplifying the first term: cos^2(theta) / (1 - tan(theta))
We know that cos^2(theta) = 1 - sin^2(theta) (from the Pythagorean identity). Thus, we can rewrite the first term as:
(1 - sin^2(theta)) / (1 - tan(theta))
Using the identity tan(theta) = sin(theta) / cos(theta), we can simplify the denominator:
(1 - sin^2(theta)) / (1 - sin(theta) / cos(theta))
Multiply both the numerator and denominator by cos(theta) to get a common denominator:
(1 - sin^2(theta)) * cos(theta) / (cos(theta) - sin(theta))
Now, expand the numerator:
(cos(theta) - sin^2(theta) * cos(theta)) / (cos(theta) - sin(theta))
2. Simplifying the second term: sin^3(theta) / (sin(theta) - cos(theta))
Since sin(theta) = cos(90 - theta), we can rewrite the term as:
(cos(90 - theta))^3 / (cos(90 - theta) - cos(theta))
Using the identity cos(90 - theta) = sin(theta), we have:
sin^3(theta) / (sin(theta) - cos(theta))
Now, we can rewrite sin^3(theta) as (sin(theta))^2 * sin(theta):
((sin(theta))^2 * sin(theta)) / (sin(theta) - cos(theta))
3. Combining the simplified terms:
Now that we have simplified both the first and second terms, we can add them together:
(cos(theta) - sin^2(theta) * cos(theta)) / (cos(theta) - sin(theta)) + ((sin(theta))^2 * sin(theta)) / (sin(theta) - cos(theta))
To combine the fractions, we need a common denominator. Multiply the first term's numerator and denominator by sin(theta):
((cos(theta) - sin^2(theta) * cos(theta)) * sin(theta)) / (sin(theta) * (cos(theta) - sin(theta)))
And multiply the second term's numerator and denominator by (cos(theta) - sin(theta)):
((sin(theta))^2 * sin(theta) * (cos(theta) - sin(theta))) / (sin(theta) * (cos(theta) - sin(theta)))
Now, we can add the two terms and simplify:
((cos(theta) - sin^2(theta) * cos(theta)) * sin(theta) + (sin(theta))^2 * sin(theta) * (cos(theta) - sin(theta))) / (sin(theta) * (cos(theta) - sin(theta)))
Expanding the equation:
cos(theta)*sin(theta) - sin^3(theta) * cos(theta) + sin^3(theta) * cos(theta) - sin^4(theta) * cos(theta)
Simplifying further:
cos(theta)*sin(theta) - sin^4(theta) * cos(theta)
Finally, factoring out cos(theta) from both terms:
cos(theta) * (sin(theta) - sin^4(theta))
And using the identity sin^2(theta) = 1 - cos^2(theta), we can rewrite the equation as:
cos(theta) * (sin(theta) - (1 - cos^2(theta))^2)
Expanding and simplifying further:
cos(theta) * (sin(theta) - (1 - 2cos^2(theta) + cos^4(theta)))
cos(theta) * (sin(theta) - 1 + 2cos^2(theta) - cos^4(theta))
cos(theta) * (2cos^2(theta) + sin(theta) - 1 - cos^4(theta))
cos(theta) * (2cos^2(theta) - cos^4(theta) + sin(theta) - 1)
The equation is not equal to 1 + sin(theta) * cos(theta), so it appears that the initial statement is not valid.
To solve the equation (cos^2θ)/(1-tanθ) + (sin^3θ)/(sinθ - cosθ) = 1 + sinθ * cosθ, we need to simplify both sides of the equation and manipulate the expressions using trigonometric identities. Let's go through the steps:
Step 1: Simplify the left side
To simplify the left side of the equation, let's start with the first term:
(cos^2θ)/(1-tanθ)
Using the identity: 1 + tan^2θ = sec^2θ, we can rewrite the denominator:
(1 - tanθ) = (sec^2θ - tan^2θ) / (sec^2θ)
Therefore, the first term becomes:
(cos^2θ) / (sec^2θ - tan^2θ)
Next, let's simplify the second term:
(sin^3θ) / (sinθ - cosθ)
Rearrange the numerator to match the denominator:
(sin^3θ) / (-cosθ + sinθ)
Step 2: Apply trigonometric identities
Using the identity: cos^2θ + sin^2θ = 1, we can rewrite the numerator of the first term:
cos^2θ = 1 - sin^2θ
Now, replace cos^2θ and sin^3θ in the equation with the corresponding identities:
(1 - sin^2θ) / (sec^2θ - tan^2θ) + sinθ / (-cosθ + sinθ)
Step 3: Simplify the right side
On the right side of the equation, we have 1 + sinθ * cosθ, which can be rearranged to:
1 + (sinθ * cosθ)
Step 4: Combine like terms
Now that we've simplified both sides, we can combine like terms on the left side of the equation. Let's find the common denominator for the two terms in the left side denominator:
(sec^2θ - tan^2θ) = (1/cos^2θ) - (sin^2θ/cos^2θ) = (1 - sin^2θ) / cos^2θ
Therefore, the left side becomes:
(1 - sin^2θ) / (1 - sin^2θ) + sinθ / (-cosθ + sinθ)
Step 5: Simplify and solve
Since the first terms in the numerator and denominator cancel out:
1 + sinθ / (-cosθ + sinθ)
The equation now simplifies to:
1 + sinθ / (-cosθ + sinθ) = 1 + sinθ * cosθ
To solve for θ, we can multiply both sides of the equation by (-cosθ + sinθ) to eliminate the denominator:
1 + sinθ = (1 + sinθ * cosθ) * (-cosθ + sinθ)
Now, expand the right side:
1 + sinθ = -cosθ - sinθ * cosθ + sin^2θ * cosθ + sin^2θ
Rearrange and group like terms:
1 + sinθ = (1 + sin^2θ) - cosθ * (1 + sinθ)
Next, distribute cosθ:
1 + sinθ = 1 + sin^2θ - cosθ - cosθ * sinθ
Combine like terms:
1 + sinθ = 1 + sin^2θ - cosθ - cosθ * sinθ
Now, subtract 1 and cosθ from both sides of the equation:
sinθ - cosθ * sinθ = sin^2θ - cosθ
Factor out sinθ on the left side:
sinθ (1 - cosθ) = sin^2θ - cosθ
Finally, divide both sides by (1 - cosθ):
sinθ = (sin^2θ - cosθ) / (1 - cosθ)
And there's the solution to the equation for θ.