if tan theta = 1/7 and tan phi= 1/3 then prove that cos 2 theta= sin 4 phi
tanθ=1/7 means sinθ=1/√50
cos 2θ = 1-2sin^2θ = 1-1/25 = 24/25
tanØ=1/3 means sinØ=1/√10 and cosØ=3/√10
sin2Ø = 2sinØcosØ = 3/5
so, cos2Ø = 4/5
sin4Ø = 2sin2Øcos2Ø = 24/25
tanθ=1/7 means sinθ=1/√50
cos 2θ = 1-2sin^2θ = 1-1/25 = 24/25
tanØ=1/3 means sinØ=1/√10 and cosØ=3/√10
sin2Ø = 2sinØcosØ = 3/5
so, cos2Ø = 4/5
sin4Ø = 2sin2Øcos2Ø = 24/25
tanθ=1/7 means sinθ=1/√50
cos 2θ = 1-2sin^2θ = 1-1/25 = 24/25
tanØ=1/3 means sinØ=1/√10 and cosØ=3/√10
sin2Ø = 2sinØcosØ = 3/5
so, cos2Ø = 4/5
sin4Ø = 2sin2Øcos2Ø = 24/25
To prove that cos 2θ = sin 4ϕ using the given information, we will need to use several trigonometric identities and algebraic manipulations.
Step 1: Recall the double-angle identity for cosine:
cos 2θ = 1 - 2sin^2θ
Step 2: Use the given value of tan θ = 1/7 to find sin θ:
Since tan θ = sin θ / cos θ, we can set up the equation:
1/7 = sin θ / cos θ
Cross multiplying, we get:
cos θ = 7sin θ
Step 3: Substitute the value of cos θ obtained in step 2 into the double-angle identity for cosine:
cos 2θ = 1 - 2sin^2θ
= 1 - 2(sin θ/cos θ)^2 (substituting 7sin θ for cos θ)
= 1 - 2(sin θ)^2 / (7sin θ)^2
= 1 - 2(sin θ)^2 / (49(sin θ)^2)
= 1 - 2 / 49
Step 4: Simplify the expression obtained in step 3:
cos 2θ = (49 - 2) / 49
= 47 / 49
Step 5: Recall the quadruple-angle identity for sine:
sin 4ϕ = 4sin ϕ cos^3ϕ
Step 6: Use the given value of tan ϕ = 1/3 to find sin ϕ:
Since tan ϕ = sin ϕ / cos ϕ, we can set up the equation:
1/3 = sin ϕ / cos ϕ
Cross multiplying, we get:
cos ϕ = 3sin ϕ
Step 7: Substitute the value of cos ϕ obtained in step 6 into the quadruple-angle identity for sine:
sin 4ϕ = 4sin ϕ cos^3ϕ
= 4sin ϕ (3sin ϕ)^3 (substituting 3sin ϕ for cos ϕ)
= 4sin ϕ * 27(sin ϕ)^3
= 108(sin ϕ)^4
Step 8: Simplify the expression obtained in step 7:
sin 4ϕ = 108(sin ϕ)^4
Comparing the expressions obtained in step 4 and step 8, we have shown that cos 2θ = sin 4ϕ using the given values of tan θ and tan ϕ.