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Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85?
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Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 1...
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Community Answer
Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 1...
Evaluation of Trigonometric Expression
Given Expression:
sec(90-θ) cosec(θ) tan(90-θ) cos² 35° cos² 55° / tan 5° tan 15° tan 45° tan 75° tan 85°
Solution:
Let's simplify the given expression using the trigonometric identities:
Step 1: Simplify the first term
sec(90-θ) = cos(θ)
Step 2: Simplify the second term
cosec(θ) = 1/sin(θ)
Step 3: Simplify the third term
tan(90-θ) = cot(θ) = 1/tan(θ)
Step 4: Simplify the numerator
cos² 35° cos² 55° = (cos 35° * cos 55°)²
Step 5: Simplify the denominator
tan 5° tan 15° tan 45° tan 75° tan 85° = sin 5° sin 15° sin 45° sin 75° sin 85°
Step 6: Substitute the simplified terms in the given expression
cos(θ) * 1/sin(θ) * 1/tan(θ) * (cos 35° * cos 55°)² / (sin 5° sin 15° sin 45° sin 75° sin 85°)
Step 7: Simplify the expression
cos 35° * cos 55° = (cos 45° + cos 20°)/2
(cos 45° + cos 20°)/2 = (√2/2 + cos 20°)/2
Now, substitute the above value in the numerator:
cos(θ) * 1/sin(θ) * 1/tan(θ) * (√2/2 + cos 20°)² / (sin 5° sin 15° sin 45° sin 75° sin 85°)
Using the identity (a+b)² = a² + 2ab + b²:
(√2/2 + cos 20°)² = 1/2 + √2 cos 20° + cos² 20°
Now, substitute the above value in the numerator:
cos(θ) * 1/sin(θ) * 1/tan(θ) * (1/2 + √2 cos 20° + cos² 20°) / (sin 5° sin 15° sin 45° sin 75° sin 85°)
Using the identity 1/sin(θ) = csc(θ) and 1/tan(θ) = cot(θ):
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Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85?
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Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85?.
Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Evaluate Sec 90-theta cosec-tan 90-theta cos^2 35 cos^2 55/tan 5 tan 15 tan 45 tan 75 tan 85?.
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