Prove each of the following identities:

(1 – cos² θ)cosec² θ = 1

Prove each of the following identities:

(1 + cot² θ) sin²θ = 1

Prove each of the following identities:

(sec² θ – 1) cot² θ = 1

Prove each of the following identities:

(sec² θ – 1)(cosec² θ – 1) = 1

Prove each of the following identities:

(1 – cos² θ) sec² θ = tan² θ

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

(1 + cos θ )(1 – cos θ)(1 + cot² θ) = 1

Prove each of the following identities:

(cosec θ)(1 + cos θ)(cosec θ – cot θ) = 1

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

sec θ(1 – sin θ)(sec θ + tan θ) = 1

Prove each of the following identities:

sin θ (1 + tan θ) + cos θ (1+ cot θ) = (sec θ + cosec θ)

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

sin⁶ θ + cos⁶ θ = 1 – 3 sin² θ cos² θ

Prove each of the following identities:

sin² θ + cos⁴θ = cos² θ + sin⁴ θ

Prove each of the following identities:

cosec⁴ θ – cosec² θ = cot⁴ θ + cot² θ

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Prove each of the following identities:

Show that none of the following is an identity:

cos² θ + cos θ = 1

Show that none of the following is an identity:

sin² θ + sin θ = 2

Show that none of the following is an identity:

tan² θ + sin θ = cos² θ

Prove that: (sin θ – 2 sin³ θ) = (2cos³ θ – cos θ) tan θ.

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that (m²+ n²) = (a² + b²).

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that

(x² – y²) = (a² – b²)

If prove that

If (sec θ + tan θ) = m and (sec θ – tan θ) = n, show that mn = 1.

If (cosec θ + cot θ) = m and (cosec θ – cot θ) = n, show that mn = 1

If x = a cos³ θ and y = b sin³ θ, prove that

If (tan θ + sin θ) = m and (tan θ – sin θ) = n, prove that (m² – n²)² = 16mn.

If (cot θ + tan θ) = m and (sec θ – cos θ) = n, prove that (m²n) ^2/3 – (mn²) ^2/3 = 1.

If (cosec θ – sin θ) = a³ and (sec θ – cos θ) = b³, prove that a² b² (a² + b²) = 1.

If (2 sin θ + 3 cos θ) = 2, prove that (3 sin θ – 2 cos θ) = ± 3.

If (sin θ + cos θ) = √2 cos θ, show that cot θ = (√2 + 1).

If (cos θ + sin θ) = √2sin θ, prove that (sin θ – cos θ) = √2cos θ.

If sec θ + tan θ = p, prove that

(i)

(ii)

(iii)

If tan A = n tan B and sin A = m sin B, prove that .

If m = (cos θ – sin θ) and n = (cos θ + sin θ) then show that

Write the value of (1 – sin²θ) sec² θ.

Write the value of (1 – cos²θ) cosec² θ.

Write the value of (1 + tan²θ) cos² θ.

Write the value of (1 + cot²θ) sin² θ.

Write the value of .

Write the value of .

Write the value of sin θ cos (90° – θ) + cos θ sin (90° – θ).

Write the value of cosec² (90° – θ) – tan² θ.

Write the value of sec² θ (1 + sin θ)(1 – sin θ).

Write the value of cosec² θ (1 + cos θ)(1 – cos θ).

Write the value of sin² θ cos² θ (1 + tan² θ)(1 + cot² θ).

Write the value of (1 + tan² θ)(1 + sin θ)(1 – sin θ).

Write the value of 3 cot² θ – 3 cosec² θ.

Write the value of .

Write the value of

If sin θ = 1/2, write the value of (3 cot² θ + 3).

If cos θ = 2/3, write the value of (4 +4 tan² θ).

If cos θ = 7/25, write the value of (tan θ + cot θ).

If cos θ = 2/3, write the value of .

If 5 tan θ = 4, write the value of .

If 3 cot θ = 4, write the value of .

If cot θ = 1/√3 write the value of

If tan θ = 1/√5write the value of

If cot A = 4/3 and (A + B) = 90°, what is the value of tan B?

If cos B = 3/5 and (A + B) = 90°, find the value of sin A.

If √3sin θ = cos θ and θ is an acute angle, find the value of θ.

Write the value of tan 10° tan 20° tan 70° tan 80°.

Write the value of tan 1° tan 2° ... tan 89°.

Write the value of cos 1° cos 2° ... cos 180°.

If tan A = 5/12, find the value of (sin A + cos A) sec A.

If sin θ = cos (θ – 45°), where θ is acute, find the value of θ.

Find the value of .

Find the value of sin 48° sec 42° + cos 48° cosec 42°.

If x = a sin θ and y = b cos θ, write the value of (b²x² + a²y²).

If 5x = sec θ and 5/x = tan θ, find the value of 5

If cosec θ = 2x and cot θ = 2/x find the value of 2.

If sec θ + tan θ = x, find the value of sec θ.

Find the value of .

If sin θ = x, write the value of cot θ.

If sec θ = x, write the value of tan θ.

tan 10° tan 15° tan 75° tan 80° = ?

tan 5° tan 25° tan 30° tan 65° tan 85° = ?

cos 1° cos 2° cos 3° ... cos 180° = ?

sin 47° cos 43° + cos 47° sin 43° = ?

sec 70° sin 20° + cos 20° cosec 70° = ?

If sin 3A = cos (A – 10°) and ∠A is acute then ∠A = ?

If sec 4A = cosec (A – 10°) and 4A is acute then ∠A = ?

If A and B are acute angles such that sin A = cos B then (A + B) = ?

If cos (α + β) = 0 then sin (α – β) = ?

sin (45° + θ) – cos (45° – θ) = ?

sec² 10° – cot² 80° = ?

cosec² 57° – tan² 33° = ?

If 2 sin 2θ = √3 then θ = ?

If 2 cos 3θ = 1 then θ = ?

If √3tan 2θ – 3 = 0 then θ = ?

If tan x = 3 cot x then x = ?

If x tan 45° cos 60° = sin 60° cot 60° then x = ?

If tan²45° – cos²30° = x sin 45° cos 45° then x = ?

sec²60° – 1 = ?

(cos 0° + sin 30° + sin 45°) (sin 90° + cos 60° – cos 45°) = ?

sin² 30° + 4 cot² 45° – sec² 60° = ?

3 cos²60° + 2 cot²30° – 5 sin²45° = ?

cos² 30° cos² 45° + 4 sec² 60° + 2cos² 90° – 2 tan² 60° = ?

If cosec θ = √10 then sec θ = ?

If tan θ = 8/15, then cosec θ = ?

If sin θ = a/b, then cos θ = ?

If tan θ = √3, then sec θ = ?

If sec θ = 25/7, then sin θ = ?

If sin θ = 1/2, then cot θ = ?

If cos θ = 4/5 then tan θ = ?

If 3x = cosec θ and 3/x = cot θ, then

If 2x = sec A and 2/x = tan A then

If tan θ = 4/3, then (sin θ + cos θ) = ?

If (tan θ + cot θ) = 5 then (tan² θ + cot² θ) = ?

If (cos θ + sec θ) = 5/2, then (cos² θ + sec² θ) = ?

If

If

If

If tan θ = a/b, then

If sin A + sin²A = 1 then cos²A + cos⁴A = ?

If cos A + cos²A = 1 then sin²A + sin⁴A = ?

If tan θ = a/b, then ?

(cosec θ – cot θ)² = ?

(sec A + tan A)(1 – sin A) = ?

The value of (sin² 30° cos² 45° + 4 tan² 30°+ (1/2) sin² 90^o + (1/8)cot² 60^o) = ?

If cos A + cos² A = 1 then (sin²A + sin⁴A) = ?

If sin θ = √3/2, then (cosec θ + cot θ) = ?

If cot A = 4/5, prove that

If 2x = sec A and 2/x = tan A, prove that .

If √3 tan θ = 3 sin θ, prove that (sin² θ – cos² θ) = 1/3.

Prove that .

If 2 sin 2θ = √3, prove that θ = 30°.

Prove that .

If cosec θ + cot θ = p, prove that .

Prove that .

If 5 cot θ = 3, show that the value of is 16/29.

Prove that (sin 32° cos 58° + cos 32° sin 58°) = 1.

If x = a sin θ + b cos θ and y = a cos θ – b sin θ, prove that x² + y² = a² + b².

Prove that .

Prove that .

Prove that .

Prove that

If sec 5A = cosec (A – 36°) and 5A is an acute angle, show that A = 21°.