Simplify ((csc^2x)(sin^4x+cos^2x))/(cos^2x) - cos^2x = tan^2x*csc^2x+cot^2x+sin^2x-1
Earlier today I gave a lengthy reply to this question which falls along the same lines as yours:
https://www.jiskha.com/questions/1829098/equation-1-tan-2x-cos-2x-sec-2x-csc-2x-sec-4x-csc-2x-cos-2x-1-equation-2
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I have a start to yours, and I will let you finish it after perusing the other post:
LS = ((csc^2x)(sin^4x+cos^2x))/(cos^2x) - cos^2x
= (1/sin^2 x)(sin^4 x + cos^2 x)/cos^2 x - cos^2 x
= (sin^2 x + cos^2 x/sin^2 x)/cos^2 x - cos^2 x
= sin^2 x/cos^2 x + 1/sin^2 x - cos^2 x
= tan^2 x + csc^2 x - cos^2 x
RS = tan^2x*csc^2x+cot^2x+sin^2x-1
= sin^2 x/cos^2 x * 1/sin^2 x + cos^2 x/ sin^2 x - 1
= 1/cos^2 x + cos^2 x /sin^2 x - 1
= see what you can from here ....
To simplify the given expression, let's break it down step by step:
Step 1: Expand and simplify the left side of the equation
((csc^2x)(sin^4x+cos^2x))/(cos^2x) - cos^2x
Using the distributive property, we can expand the numerator:
(csc^2x * sin^4x + csc^2x * cos^2x) / (cos^2x) - cos^2x
Step 2: Simplify the first term in the numerator
The first term, csc^2x * sin^4x, can be simplified using the identity csc^2x = 1/sin^2x:
(1/sin^2x * sin^4x + csc^2x * cos^2x) / (cos^2x) - cos^2x
Simplifying further:
sin^2x + csc^2x * cos^2x / (cos^2x) - cos^2x
Step 3: Simplify the second term in the numerator
The second term, csc^2x * cos^2x / (cos^2x), simplifies to just csc^2x:
sin^2x + csc^2x - cos^2x
Step 4: Simplify the denominator
The denominator remains the same:
cos^2x
Step 5: Rearrange the terms
Combine like terms in the numerator:
(sin^2x - cos^2x) + csc^2x
Step 6: Apply trigonometric identities
Using the Pythagorean identity sin^2x = 1 - cos^2x, we can substitute it into the expression:
(1 - cos^2x - cos^2x) + csc^2x
Simplifying further:
1 - 2cos^2x + csc^2x
Step 7: Apply more trigonometric identities
Using the identity csc^2x = 1 + cot^2x, we can substitute it into the expression:
1 - 2cos^2x + (1 + cot^2x)
Simplifying further:
2 - 2cos^2x + cot^2x
Step 8: Apply the identity cot^2x = 1 + tan^2x
Substituting that into the expression:
2 - 2cos^2x + (1 + tan^2x)
Simplifying further:
3 - 2cos^2x + tan^2x
Step 9: Simplify the expression
Finally, the simplified expression is:
tan^2x - 2cos^2x + 3